a lecture course on the atiyah-singer index theorem · a lecture course on the atiyah-singer index...

A LECTURE COURSE ON THE ATIYAH-SINGER INDEX

THEOREM

JOHANNES EBERT

Abstract. These lecture notes arose from a two-semester course that I gave

in the academic year 2013/14. The goal that was achieved during the courseis a complete proof of the Atiyah-Singer index theorem for Dirac operators.

Thanks go to the students who attended the course; in particular to Matthias

Wink and Paul Breutmann for reading substantial parts of this manuscript.This is not the final version of these notes, a substantial revision will be

done. In particular, I have not included proper references throughout, espe-

cially the bibliography for the second part contains many gaps.

Contents

1. Index theory in abstract functional analysis 41.1. Generalities on Fredholm operators and the statement of the Toeplitz

index theorem 41.2. Some functional analysis 1: the open mapping theorem and its

consequences 61.3. Some functional analysis 2: the adjoint operator 71.4. Some functional analysis 3: Compact operators 91.5. Atkinsons Lemma and its consequences 111.6. Proof of the Toeplitz index theorem 121.7. A generalization 131.8. An example from ordinary differential equations 151.9. Literature and remarks 172. Differential operators on manifolds and the de Rham complex revisited 182.1. The de Rham operator 182.2. Differential operators in general 192.3. The formal adjoint 262.4. Literature 303. Analysis of elliptic operators 313.1. Preliminaries: Convolution and Fourier transformation 313.2. Sobolev spaces in Rn 353.3. The fundamental elliptic estimate 383.4. A smoothing procedure 423.5. Local elliptic regularity 463.6. Sobolev spaces on manifolds 463.7. Global regularity and the Hodge theorem 483.8. The spectral decomposition of an elliptic operator 513.9. Guide to the literature 534. Some interesting examples of differential operators 54

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2 JOHANNES EBERT

4.1. The Euler number 544.2. The signature 554.3. Complex manifolds and vector bundles 584.4. Multilinear algebra of complex vector spaces 594.5. The Dolbeault (Cauchy-Riemann) operator 614.6. The Hodge decomposition on a Riemann surface 624.7. Relation to the classical theory 654.8. Literature 675. Some bundle techniques 685.1. Vector bundles 685.2. Principal bundles 725.3. Literature 776. More on de Rham cohomology 786.1. Technical prelimiaries 786.2. Poincare duality - again! -, Kunneth theorem and the Thom

isomorphism 816.3. Geometric interpretation of the Thom class and the Poincare-Hopf

theorem 876.4. The topological Gauß-Bonnet theorem 916.5. The Gysin map and the splitting principle for complex vector bundles 936.6. Literature 957. Connections, curvature and the Chern-Weil construction 967.1. Covariant derivatives 967.2. The first Chern class - the baby case of Chern-Weil theory 977.3. The coordinate description of a connection 997.4. The very basics of Lie theory 1017.5. Back to connections 1047.6. Linear connections induced by a principal connection 1067.7. The Chern-Weil construction 1077.8. Chern classes and the proof of the Riemann-Roch theorem 1107.9. Proof of the Gauß-Bonnet-Chern theorem 1147.10. Remarks 1178. Complex K-theory 1198.1. Recapitulation of the definition 1198.2. Clutching constructions for vector bundles 1208.3. The exact sequence 1218.4. Relative K-theory 1248.5. K-theory with compact support and the statement of the Atiyah-Singer

index theorem 1318.6. Proof of Bott periodicity - the formal part 1348.7. The index bundle of a Fredholm family 1368.8. Historical remarks and references 1419. Advanced theory of characteristic classes 1429.1. Recapitulation of the Chern-Weil construction 1429.2. The universal perspective 1439.3. Chern, Euler and Pontrjagin classes: the invariants of the classical

groups 1459.4. Maximal tori in compact Lie groups 153

A LECTURE COURSE ON THE ATIYAH-SINGER INDEX THEOREM 3

9.5. Proof of Cartan’s theorem 1559.6. Examples 1579.7. The Chern character 1629.8. Chern character and Thom isomorphisms 1679.9. The cohomological version of the index theorem 1689.10. universal symbols 1709.11. The classical index theorems 1729.12. Literature 17610. Dirac operators and proof of the index theorem: the easy part 17710.1. The definition 17710.2. The classical operators are Dirac operators 17810.3. Clifford bundles 17910.4. Algebra 18010.5. The Atiyah-Bott-Shapiro construction 18210.6. The bundle modification theorem 18510.7. Representation theory of Clifford algebras - algebraic Bott periodicity18910.8. Spinc-manifolds and the index theorem on the sphere 19210.9. The bordism theorem 19410.10. The reformulation of the bordism theorem 19710.11. Elliptic analysis on manifolds with cylindrical ends 19810.12. The proof of the index theorem 20410.13. Guide to the literature 208References 209

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1. Index theory in abstract functional analysis

The meaning of the word ”abstract” is that we consider operators on abstractHilbert spaces, not differential operators. This chapter is intended as a warm-up toindex theory. Besides reviewing some of the basic principles from linear functionalanalysis and learning the definition of a Fredholm operator, we will prove the firsttheorem of this course: the Toeplitz index theorem. To each map f ∶ S1 → C×, wewill define a Fredholm operator Tf whose index is −deg(f), the classical windingnumber. Thus we see that one of the most basic topological invariants have a niceinterpretation as an index. This will be an important ingredient of the proof of theBott periodicity theorem, which in turn is fundamental for the Atiyah-Singer indextheorem.

1.1. Generalities on Fredholm operators and the statement of the Toeplitzindex theorem.

Definition 1.1.1. Let V and W be two vector spaces (usually over C). A linearmap F ∶ V →W is called a Fredholm operator if ker(F ) and coker(F ) ∶=W / Im(F )are both finite-dimensional. The index of F is by definition ind(F ) ∶= dim ker(F )−dim coker(F ).

Lemma 1.1.2. If V and W are finite dimensional vector spaces, then any linearmap F ∶ V →W is Fredholm and its index is ind(F ) ∶= dim(V ) − dim(W ).

Proof. Recall the rank-nullity theorem from Linear Algebra I; it says that dim Im(F ) =dim(V )−dim ker(F ). Thus ind(F ) = (dim(V )−dim Im(F ))−(dim(W )−dim Im(F )) =dim(V ) − dim(W ).

Lemma 1.1.3. If UG→ V

F→ W be two linear maps. If two of the three operatorsG, F , F G are Fredholm, then so is the third, and

ind(F G) = ind(F ) + ind(G).

Proof. There is a commutative diagram

(1.1.4) 0 // U(id,G) //

G

U ⊕ V −G+id //

(FG,id)

V

F

// 0

0 // V(F,id) // W ⊕ V −id+F // W // 0

and both rows are exact sequences. Now we view the columns as chain complexesand get a six-term exact sequence

0→ ker(G)→ ker(FG)→ ker(F )→ coker(G)→ coker(FG)→ coker(F )→ 0,

using that ker(FG) ≅ ker(FG⊕ id), and the analogous relation for the cokernels.This is the (not very) long exact homology sequence of the short exact sequence1.1.4 of chain complexes. Now an exercise in linear algebra shows:

dim ker(G)−dim ker(FG)+dim ker(F )−dim coker(G)+dim coker(FG)−dim coker(F ) = 0,

which is what we wanted to show.


Exercise 1.1.5. If you do not understand how the exact sequence arose in theabove proof, take a homological algebra book and read the section on the long exacthomology sequence. Do the linear algebra exercise.

This bourbakian approach cannot be pursued much longer: by means of purelinear algebra, we cannot say more on indices of operators. In the sequel, we willonly study continuous linear maps of Banach spaces, in fact, only of Hilbert spaces.We have to recall some notions and results from basic functional analysis. Considera vector space V over C, together with a scalar product V ×V → C, (x, y)↦ (x, y).The scalar product is C-sesquilinear and positive definite.

We define the norm induced by the scalar product by ∥x∥ ∶=√

(x,x). V is calleda Hilbert space if the norm is complete, i.e. if each Cauchy sequence converges.

Lemma 1.1.6. A linear map f ∶ V → W of normed vector spaces is continuousif and only if there is a C ≥ 0 with ∥f(x)∥ ≤ C∥x∥ for all x. The smallest such Cis called the operator norm ∥f∥. An alternative word for continuous linear map is”bounded operator”, and Lin(V ;W ) is the set of bounded linear maps.

This is Lemma 5.6 in [38]. An important class of operators on a Hilbert space arethe projection operators. Let V be a Hilbert space and U ⊂ V be a closed subspace.By U, we denote the orthogonal complement of U in V . Any element v ∈ V canbe written uniquely as v = Pv + (v −PV ), Pv ∈ U , (v −Pv) ∈ U. The map v ↦ Pvis the projection operator. It has the properties P 2 = P and Im(P ) = U . Moreover,∥P ∥ = 1 if 0 ≠ U .

Exercise 1.1.7. Show that Lin(V,W ), together with the operator norm, is a normedvector space. Prove that ∥fg∥ ≤ ∥f∥∥g∥. Prove that Lin(V ;W ) is complete if W iscomplete. Is it a Hilbert space?

The archetypical Hilbert space is L2(X;µ) for a measure space (X,µ). Specialcases: X = S a discrete set and µ the counting measure. In that case, we call it`2(S). The shift operator T− ∶ `2(N) → `2(N) is defined by setting T−ei ∶= ei+1,where ei is the canonical ith basis vector. It is bounded with norm 1 and is aFredholm operator with index −1. There is another shift operator T+ ∶ `2 → `2,T+ei ∶= ei−1 and T+e1 = 0: It has index 1.

This is not a linear algebra class; we want to geometrize these examples. Let uslook at the space S1 = z ∈ C∣∣z∣ = 1. This is a Riemann manifold with volume form

12πi

dzz

. We look at the space L2(S1) of complex valued square integrable functions

S1 → C; the scalar product is given by

(f, g) ∶= 1

2πi∫S1fgdz

z.

An orthonormal basis is given by the functions fk(z) = zk, k ∈ Z. By meansof this basis, we identify L2(S1) with `2(Z) (Fourier series!). You might it findmore convenient to identify L2(S1) with the space of all 1-periodic functions on

R; the scalar product has the alternative form ∫1

0 fgdx, the above orthonormal

basis corresponds to e2πikx. You are mathematically mature and should not try toseparate real and imaginary part of a function.

Inside L2(S1), we find an important subspace H(S1); it is the closure of thelinear span of all the functions fk with k ≥ 0. The space H(S1) is also called theHardy space. There is a linear orthogonal projection operator P ∶ L2(S1)→H(S1).

6 JOHANNES EBERT

Note that by a standard abuse of terminology, a projection P is called ”orthogonal”if it is selfadjoint (see below). Under the above isometry L2(S1) ≅ `2(Z), thesubspace H(S1) corresponds to `2(N).

Another important operator is given when f ∶ S1 → C is a continuous function;it sends u ∈ L2(S1) to Mfu ∶= fu. This is a bounded operator with ∥Mf∥ ≤ ∥f∥C0 .

Definition 1.1.8. Let f ∶ S1 → C be a continuous function. The Toeplitz operatorTf ∶H(S1)→H(S1) is given by Tfu ∶= PMfu.

Note that

(1.1.9) ∥Tfx∥ ≤ ∥f∥C0∥x∥.Therefore, Tf is bounded and ∥Tf∥ ≤ ∥f∥C0 ; and so Tf depends continuously on

f . Example: if f(z) = z±1, then Tf is the shift T∓. More generally, one can consider

powers of these operators; for example, Tzk = (T−)k if k ≥ 0, but not if k < 0.What can we say about continuous maps f ∶ S1 → C×? There is an important

topological invariant, the winding number or mapping degree. We have the following(equivalent) definitions, see ”Manifolds and differential forms” and ”Topology I”.

The fundamental group π1(C×) is isomorphic to Z via the isomorphism ψ ∶Z→ π1(C×), which sends the number n to the (homotopy class of the) closedloop t↦ e2πint. If f ∶ S1 → C× is any map, the closed loop t↦ f(e2πit)/f(1)represents an element [[f]] ∈ π1(C×), and we put deg(f) ∶= ψ−1([[f]]).

Any map S1 → S1 induces a self-map of the first homology groupH1(S1;Z) ≅Z; it is given by mutiplication with an integer n.

Assume that f is smooth, and consider a regular value z of the functiong = f

∣f ∣ ∶ S1 → S1 and count preimages g−1(z) with sign (the sign is the sign

of the derivative of g). If f is not smooth, take a smooth approximation. deg(f) ∶= ∫S1 f

∗( dz2πiz

) if f is smooth.

Remark 1.1.10. The maps π1(C×)→ [S1;C×] deg→ Z (the first one is the most ob-vious one) are both isomorphisms. This is notable since the group structures in thefirst two groups have two sources: in the fundamental group, you take compositionof loops, in the second one, the pointwise product of functions. The second set hasa group structure since C× is a topological (even Lie) group.

From these considerations, we see that for fk(z) = zk, ind(Tfk) = −deg(fk) = −k.The first real theorem of this lecture course is

Theorem 1.1.11. (The Toeplitz index theorem) If f ∶ S1 → C× is continuous, thenTf is a Fredholm operator and ind(Tf) = −deg(f).

Note that we just proved the Toeplitz index theorem for the special functions fk.A concrete description of Tf as a infinite matrix is not available and not practical,we need more clever tools. The first thing we need is a general principle to provethat an operator is Fredholm. This means, we have to absorb a crash course onsome parts of functional analysis.

1.2. Some functional analysis 1: the open mapping theorem and its con-sequences. A basic reference that contains (almost) all the abstract functionalanalysis we need is Hirzebruch-Scharlau, ”Einfuhrung in die Funktionalanalysis”[38]. You should have a copy on your desk.


The first thing we recall is the open mapping theorem.

Theorem 1.2.1. (The open mapping theorem) Let V and W be two Banach spacesand let F ∶ V → W be a continuous linear map. If F is surjective, then F is anopen map (i.e, images of open sets in V are open in W ).

This is Satz 9.1 in [38].

Exercise 1.2.2. Read the proof of Theorem 1.2.1.

Exercise 1.2.3. Give counterexamples that show that the completeness of bothspaces is essential in the theorem.

The converse of the open mapping theorem (”an open linear map is surjective”)is easy (why?). The most important consequence of the open mapping theorem is:

Corollary 1.2.4. A bijective continuous linear map F of Banach spaces is a home-omorphism. In particular, the inverse F −1 is bounded as well.

Lemma 1.2.5. If F ∶ V → W is Fredholm, the image F (V ) ⊂ W is a closedsubspace.

Proof. Let U ⊂W be a complement of f(V ). Since U is finite-dimensional, there is,up to equivalence, exactly one norm on U (Analysis II). The operator F1 ∶ V ⊕U →W , (v, u)↦ F (v)+u, is surjective and bounded, hence an open map by 1.2.1. Thesubset V ⊕U ∖ V ⊕ 0 is open and so is F1(V ⊕U ∖ V ⊕ 0) =W ∖ F (V ).

We denote by Lin(V,W )× ⊂ Lin(V,W ) the subset of all invertible operators.

Proposition 1.2.6. The subset Lin(V,W )× ⊂ Lin(V,W ) is open and the inversionmap F ↦ F −1 is continuous. More precisely, if F ∈ Lin×(V,W ) and R ∈ Lin(V,W )with ∥R∥ < ∥F −1∥−1, then F −R ∈ Lin×(V,W ).

Proof. (compare [38], 23.2.) The geometric series F −1∑∞k=0(RF −1)k converges to

(F −R)−1.

Theorem 1.2.7. (Homotopy invariance of the index) Let V , W be Hilbert spacesand let Fred(V,W ) be the set of all Fredholm operators. Then Fred(V,W ) ⊂Lin(V,W ) is an open subset and the index function ind ∶ Fred(V,W ) → Z, F ↦ind(F ) is locally constant.

Proof. Let F ∈ Fred(V,W ). Let G ∶ ker(F ) → V be the inclusion and H ∶ W →Im(F ) be the orthogonal projection which exists by Lemma 1.2.5 and because Wis a Hilbert space. These two are Fredholm operators with ind(G) = −dim(ker(F ))and ind(H) = dim coker(F ). The composition HFG is invertible. By Proposition1.2.6, we get that for all F1 sufficiently close to F , the composition HF1G is in-vertible. Thus HF1G and H are Fredholm, and so is F1G by Lemma 1.1.3; andind(F1G) + dim coker(F ) = ind(F1G) + ind(H) = 0.

Again by Lemma 1.1.3, F1 is Fredholm and ind(F1) = ind(F1G) − ind(G) =−dim coker(F ) + dim ker(F ) = ind(F ).

1.3. Some functional analysis 2: the adjoint operator. Let V be a normedvector space and V ′ the dual space, i.e., the vector space of all continuous linearfunctions V → C. This is a normed vector space, and moreover complete, since Cis complete.

8 JOHANNES EBERT

Definition 1.3.1. Let F ∶ V → W be a linear continuous map. The transposeoperator is F ′ ∶W ′ → V ′, F ′(φ)(v) ∶= φ(F (v)).

This has some obvious properties (linearity, (FG)′ = G′F ′, etc) which we willnot recall. It is not absolutely clear that ∥F ′∥ = ∥F ∥. That ∥F ′∥ ≤ ∥F ∥ follows fromthe definitions, and ∥F ∥ ≤ ∥F ′∥ follows from the Hahn-Banach theorem.

A special property of Hilbert spaces is that they are self-dual:

Proposition 1.3.2. Let V be a Hilbert space. Then the C-antilinear map V → V ′,v ↦ ⟨v, ⟩ is an isometry.

This is Satz 20.9 in [38]. The following lemma will be used only much later, inthe discussion of Sobolev spaces, but fits thematically.

Lemma 1.3.3. Let W , V be Hilbert spaces and F ∶W → V ′ a C-linear or antilinearbounded operator. Suppose there exists C,C ′ such that

∥v∥ ≤ C supw∈W,∣w∣≤1

∣F (w)(v)∣

and∥w∥ ≤ C ′ sup

v∈V,∣v∣≤1

∣F (w)(v)∣

hold for all v ∈ V , w ∈W . Then F is bijective (and hence a homeomorphism)

Proof. By the definition of the various norms and the assumption, we have, by thesecond estimate,

∥Fw∥ = supw∈W,∣w∣≤1

∣F (w)(v)∣ ≥ 1

C∥w∥.

Therefore F is injective (clear), and the image of F is closed, because if Fwn → v′,then ∥wn −wm∥ ≤ C∥Fwn − Fwm∥→ 0 and wn is a Cauchy sequence.

For the surjectivity, assume that v′′ ∈ V ′′ is a linear form such that v′′ F = 0.We have to prove that v′′ = 0. By duality, v′′ is given by scalar product with anv ∈ V : v′′(v′) = ⟨v, v′⟩. We knwow that F (w)(v) = 0 for all w ∈ W . Therefore, bythe first estimate in the assumption of the Lemma, ∥v∥ = 0, as claimed.

Using Proposition 1.3.2, we can define the adjoint operator F ∗ of a linear oper-ator F ∶ V →W .

Definition 1.3.4. Let F ∶ V → W be a bounded operator of Hilbert spaces. The

adjoint F ∗ of F is the composition W ≅W ′ F′

→ V ′ ≅ V .

Proposition 1.3.5.

(1) F ↦ F ∗ is antilinear.(2) F ∗∗ = F .(3) (FG)∗ = G∗F ∗.(4) ⟨Fv,w⟩ = ⟨v,F ∗w⟩.(5) ∥F ∥ = ∥F ∗∥.(6) ∥F ∗F ∥ = ∥F ∥2.

Proof. Everything is clear except perhaps the last equation. For each v,w ∈ V ofnorm ≤ 1, we have

∣⟨v,F ∗Fw⟩∣ = ∣⟨Fv,Fw⟩∣ ≤ ∥F ∥2


and

∥Fv∥2 = ∣⟨Fv,Fv⟩∣ = ∣⟨v,F ∗Fv⟩∣ ≤ ∥F ∗F ∥.

Moreover, we often need the following relations.

Proposition 1.3.6. Let V,W be Hilbert spaces and F ∶ V → W be a boundedoperator. Then

(1) ker(F ∗) = ImF

(2) ImF = ker(F ∗).

Note that taking the closure is necessary for the second relation.

1.4. Some functional analysis 3: Compact operators.

Definition 1.4.1. Let V,W be Banach spaces. A bounded operator F ∶ V →W iscalled compact if one of the following equivalent conditions hold.

(1) The image of each bounded X ⊂ V is relatively compact in W .(2) The image of the open unit ball B1(V ) is relatively compact in W .(3) If vn is a bounded sequence, then Fvn does have a convergent subsequence.

We denote by Kom(V ;W ) ⊂ Lin(V,W ) the subset of all compact operators

Let us recall some well-known equivalent formulations of compactness for a met-ric space X. We say that X is totally bounded if for each ε > 0, there exist finitelymany x1, . . . , xn ∈X such that the ε-balls around the xi’s cover X, i.e.

n

⋃i=1

Dε(xi) =X.

A metric space X is compact if and only if it is complete and totally bounded.

Examples 1.4.2.Each operator with a finite-dimensional image is compact.The identity on V is compact iff V is finite dimensional.

Proof. The first follows from the Heine-Borel theorem, as well as one half of thesecond statement. For the converse, see Lemma 24.2 [38].

Theorem 1.4.3. Let V,W,U,X be Banach spaces. Then:

(1) Kom(V,W ) ⊂ Lin(V,W ) is a closed subspace.(2) If F ∈ Kom(V,W ) and G ∈ Lin(W,X); H ∈ Lin(U ;V ), then GFH ∈

Kom(U,X).(3) If F ∈ Kom(V,W ), then F ′ ∈ Kom(W ′, V ′).

Proof. (Compare [38], Lemma 24.3)Part (2): Let A ⊂ U be bounded. Then H(A) ⊂ V is bounded and FH(A) ⊂W

relatively compact. Thus FH(A) ⊂ W and hence GFH(A) ⊂ X is compact. But

GFH(A) ⊂ GFH(A) = GFH(A) is compact.Part (1): If F is compact, then clearly so is aF , a ∈ C. Moreover, if F and

G are compact, then F ⊕ G is compact as an operator V ⊕ V → W ⊕W . Thediagonal ∆ ∶ V → V ⊕ V and the sum µ ∶ W ⊕W → W are bounded, and thusµ (F ⊕G) ∆ = F +G is compact by part (2). So Kom(V,W ) ⊂ Lin(V,W ) is a

10 JOHANNES EBERT

subspace. To prove that it is closed, assume that F ∈ Kom(V,W ) and let ε > 0.Pick G ∈ Kom(V,W ) with ∥F −G∥ < ε/3.

Then G(B1(V )) can be covered by finitely many ε/3-balls around Gv1, . . . ,Gvnsince it is relatively compact. Then for each v ∈ B1(V ), there exists an i such that∥Gv −Gvi∥ < ε/3 and therefore

∥Fv − Fvi∥ ≤ ∥Fv −Gv∥ + ∥Gv −Gvi∥ + ∥Gvi − Fvi∥ < ε;therefore F (B1(V )) can be covered by finitely many ε-balls. Since ε was arbi-

trary, the set F (B1(V )) is relatively compact.The third statement uses the Arzela-Ascoli theorem, which we first state.

Theorem 1.4.4. (Arzela-Ascoli theorem) Let K be a topological space and X be acomplete metric space. A set A ⊂ C(K,X) of continuous functions K →X is calledequicontinuous if for all ε > 0 and each y ∈K, there exists a neighborhood U ⊂K ofy such that for all f ∈ A and z ∈ U , one has d(f(y), f(z)) < ε. Let A ⊂ C(K,X) beequicontinuous.

(1) If K is compact and if for all y ∈ K, the set Ay = f(y)∣f ∈ A ⊂ X isrelatively compact, then A ⊂ C(K,X) is relatively compact, where C(K,X)carries the metric d(f, g) = supy∈K d(f(y), g(y)).

(2) If K has a countable dense subset S such that for each y ∈ S, the set Ayis relatively compact, then any sequence fn has a subsequence fnk whichconverges uniformly on each compact subset of K.

The first part is proven in [38], Korollar 3.1. The second part (which is verysimilar), is Theorem 11.28 in [59].

Example 1.4.5. Let K ⊂ V be any subset of a Banach space and A ⊂ Lin(V,W )be bounded. Then A ⊂ C(K,W ) is equicontinuous.

Proof. Let x, z ∈ K and F ∈ A. Then ∥Fz − Fx∥ ≤ ∥F ∥∥z − x∥ ≤ C∥z − x∥ with C aglobal bound on A. Then U =Dε/2C is the desired neighborhood.

Proof of Theorem 1.4.3 (3). Let X ⊂ W ′ be bounded. Then X ∣F (B1(V )) is rela-

tively compact, by Arzela-Ascoli and since F is compact. Thus if (`n)n ⊂ W ′ isa bounded sequence, then there is a subsequence `nk such that `nk converges uni-

formly on F (B1(V )). Hence `nk F = F ′(`nk) converges uniformly on B1(V ).

In the Hilbert space setting, we can phrase Theorem 1.4.3 by saying that Kom(V ) ⊂Lin(V ) is a 2-sided ∗-ideal in the C∗-algebra Lin(V ).Proposition 1.4.6. Let V,W be separable Hilbert spaces. Then Kom(V,W ) is theclosure of the space of finite rank operators.

Before giving the proof, we introduce terminology. We say that a sequence xn ina metric space X is subconvergent if xn has a convergent subsequence. In argumentsthat involve picking a subsequence, we will also often denote the subsequence byxn as well, instead of using stacked indices such as xnk or worse.

Proof. Let F be a compact operator. Pick an orthonormal basis (en)n∈N of W andlet Pn ∶W →W be the orthogonal projection operator onto spanei∣i ≤ n. ClearlyPnF has finite rank, and we claim that PnF subconverges to F .

Let K = F (B1(V )), which by assumption is compact. Consider the sequencePn∣K of functions K → W . The family Pn∣K is equicontinuous because it is


bounded (Example 1.4.5). Moreover, for each x ∈K, the sequence Pnx converges tox, which is why Pnx is relatively compact in W . Thus the Arzela-Ascoli theoremapplies and proves that the family Pn∣K is relatively compact; thus a subsequenceof Pn∣K is uniformly convergent. Therefore, PnF is uniformly subconvergent onB1(V ), and the pointwise limit is F , and so PnF → F , which is why F is in theclosure of the finite-rank operators.

1.5. Atkinsons Lemma and its consequences. The next result is our main toolto prove that a given operator is Fredholm. We now restrict to separable Hilbertspaces. The exposition follows [36] and [15].

Theorem 1.5.1. (Atkinson’s Lemma) Let V,W be separable Hilbert spaces andF ∈ Lin(V,W ). Then F is Fredholm if and only if there exists G ∈ Lin(W,V ) suchthat GF − 1and FG − 1 are compact. Such an operator G is called parametrix.

Proof. Assume first that F is Fredholm. The operator

F0 ∶ ker(F ) → Im(F )is a bijective operator of Hilbert spaces (by 1.2.5, the target is complete) and

thus its inverse is bounded by the open mapping theorem 1.2.1. Let G be the

composition W → Im(F )F−1

0→ ker(F ) ⊂ V . It is easy to see that FG−1 and GF −1have finite rank and are thus compact.

For the converse direction, let GF = 1 +K and FG = 1 + L with compact K,L.Chose finite rank operators R,S with ∥R −K∥, ∥S − L∥ < 1, by Proposition 1.4.6.Then 1−R+K and 1−S+L are invertible by Proposition 1.2.6. Now compute that

(1 −R +K)−1GF = (1 −R +K)−1(1 −R +K +R) = 1 + (1 −R +K)−1R =∶ 1 + Pwith P an operator of finite rank. Thus if Fv = 0, then v +Pv = 0, i.e. ker(F ) ⊂

Im(P ); in particular, the kernel of F is finite-dimensional. On the other hand,

FG(1 − S +L)−1 = (1 − S +L + S)(1 − S +L)−1 = 1 + S(1 − S +L)−1 =∶ 1 +Qwith Q of finite rank. Thus Im(1 +Q) ⊂ Im(F ). But ker(Q) ⊂ Im(1 +Q), and

since Q has finite rank, ker(Q) has finite codimension, and therefore Im(F ) hasfinite codimension as well.

Corollary 1.5.2. Let F ∈ Fred(V,W ) and K ∈ Kom(V,W ). Then

(1) F +K ∈ Fred(V,W ) and ind(F +K) = ind(F ).(2) A self-adjoint Fredholm operator has index 0.(3) F ∗ ∈ Fred(W,V ) and ind(F ∗) = − ind(F ).

Proof. Part (1): Let G be a parametrix for F . Then G(F +K)−1 = GF −1+GK iscompact; similarly, (F +K)G − 1 is compact and one can apply Atkinson’s lemmato see that F +K is Fredholm. For each t ∈ [0,1], the operator tK is also compact,and thus ind(F + tK) does not not depend on t, by Theorem 1.2.7.Part (2): Let F be self-adjoint and Fredholm. Then Im(F ) is closed, and Im(F ) =ker(F ∗) = ker(F ). Thus the index is zero.Part (3): If G is a parametrix for F , then G∗ is a parametrix for F ∗, showing thatF ∗ is Fredholm. To compute the index, we consider the operator F ∗F which isself-adjoint and thus has index 0. Therefore ind(F ) + ind(F ∗) = 0.

12 JOHANNES EBERT

1.6. Proof of the Toeplitz index theorem. We now have amassed enoughknowledge to prove the Toeplitz index theorem quite easily. Recall that P is theprojection onto H(S1) ⊂ L2(S1;C). It turns out that it is formally simpler to con-sider the operator is PMfP + (1 − P ), as an operator L2(S1) to itself. This is thedirect sum of the Toeplitz operator and the identity and thus has the same index.In particular, we redefine Tf ∶= PMfP + (1 − P ).

Lemma 1.6.1. For each f ∈ C0(S1), the operator [P,Mf ] ∶= PMf − MfP iscompact.

Proof. Let A ⊂ C0(S1;C) the subset of all f such that PMf −MfP is compact.We verify the hypotheses of the Stone-Weierstraß theorem 1.6.2 in order to showthat A = C0(S1;C). Let

Φ ∶ C0(S1;C)→ Lin(L2(S1)); f ↦ [P,Mf ].This is a continuous linear map because ∥[P,Mf ]∥ = ∥PMf −MfP ∥ ≤ 2∥f∥C0

and hence A ∶= Φ−1(Kom(L2(S1))) is a closed subspace by Theorem 1.4.3 (1).It is clear that the constant function f = 1 is in A. If f, g ∈ A, then

PMfg −MfgP = PMfMg −MfMgP = PMfMg −MfPMg +MfPMg −MfMgP =[P ;Mf ]Mg +Mf [P,Mg];

and this is compact, so fg ∈ A. If f ∈ A, then

PMf −MfP = [Mf , P ]∗

because P = P ∗ and Mf = M∗f . For the function f(z) = z, direct computation

shows that

(PMz −MzP )(zk) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0 k < −1

1 k = −1

0 k ≥ 0

and therefore PMz −MzP has finite rank. Apply the Stone-Weierstraß theorem.

Theorem 1.6.2. (The Stone-Weierstrass theorem) Let X be a compact Hausdorffspace and A ⊂ C0(X,C). Assume

(1) A is a closed subalgebra,(2) 1 ∈ A,(3) f ∈ A⇒ f ∈ A,(4) for all x ≠ y ∈X, there is f ∈ A with f(x) ≠ f(y).

Then A = C0(X,C).

The proof can be found in [46], Theorem III.1.4.

Lemma 1.6.3. Tfg − TfTg is compact.

Proof.Tfg − TfTg = P [Mf , P ]MgP.

Corollary 1.6.4. If f ∶ S1 → C× is continuous, then Tf is Fredholm.


Proof. TfTf−1 − 1 and Tf−1Tf − 1 are compact by Lemma 1.6.3. Apply Atkinson’stheorem.

Corollary 1.6.5. If f, g ∶ S1 → C×, then ind(Tfg) = ind(Tf) + ind(Tg).

Proof. This follows from Lemma 1.6.3, 1.5.2 (1) and 1.1.3.

Proof of the Toeplitz index formula. We know, from Topology I, that f is homo-topic to zk, for k = deg(f). Since ∥Tf∥ ≤ ∥f∥, the map C0(S1;C)→ Lin(L2(S1), L2(S1)),f ↦ Tf , is continuous. Therefore, by Theorem 1.2.7, the index of Tf is equal toind(Tzk).

Since Tfg = TfTg + r for a compact r, we see that ind(Tfg) = ind(Tf) + ind(Tg).Thus ind(Tzk) = k ind(Tz), but this index was computed directly.

Exercise 1.6.6. We did not use that P is the projection onto H that often. Prove:if Q is another orthogonal projection such that P −Q is compact, then the operatorQMfQ + (1 −Q) is Fredholm. What is its index?

1.7. A generalization. We now take a slightly more abstract viewpoint on theproof of the Toeplitz index theorem. This will be the germ of the Bott period-icity theorem. We have shown that [S1,C×] → Z, f ↦ ind(Tf) is a well-definedhomomorphism (the source has a group structure by pointwise multiplication offunctions), by Corollary 1.6.5.

Moreover, recall that the natural map π1(C×)→ [S1,C×] is a group isomorphism.This is easy, but has some content; composition in the fundamental group is definedby concatenation of paths, and in the group of free homotopy classes, it is definedby multiplication.

Thus a different version of the Toeplitz index theorem is that

π1(S1)→ Z; [f]↦ ind(Tf)is an isomorphism that is equal to minus the degree.Now we define Toeplitz operators to matrix-valued functions S1 →Matn,n(C) =∶

C(n). Let H(S1)n ⊂ L2(S1;Cn) be the space spanned of all functions whose Fouriercoefficients with negative indices are zero and let Pn be the orthogonal projectiononto H(S1)n. Of course, we could write Pn as an n × n-matrix of linear operators

Pn =⎛⎜⎝

P . . . . . .. . . P . . .. . . . . . P

⎞⎟⎠.

For any matrix valued function f ∶ S1 → C(n), we can form

Tf ∶= PnMfPn + (1 − Pn).For two such functions f, g, we compute that TfTg−Tfg is the matrix of operators

whose (i, k)-entry is

n

∑j=1

(PfijPgjkP − PfijgjkP ) ≡ Pn

∑j=1

[fij ;P ]gjk (mod Kom),

which is compact by Lemma 1.6.1. This proves:

Lemma 1.7.1. If f ∶ S1 → GLn(C) is continuous, then Tf is Fredholm. If f, g aretwo such functions, then ind(Tfg) = ind(Tf) + ind(Tg).

14 JOHANNES EBERT

Now if G is any Lie group (here GLn(C)), then π1(G) is abelian and π1(G) →[S1;G] is an isomorphism.

Lemma 1.7.2. If n ≥ 1, then π1(GLn(C)) ≅ Z. More precisely, the inclusionGL1(C) → GLn(C), z ↦ diag(z,1, . . . ,1) and the determinant GLn(C) → C× =GL1(C) induce mutually inverse isomorphisms.

Proof. Observe that the maps

GLn(C)→ SLn(C) ×GL1(C); A↦ (diag(detA−1,1, . . . ,1)A; det(A))and

SLn(C) ×GL1(C)→ GLn(C); (B, z)↦ diag(z,1, . . . ,1)Bare mutually inverse homeomorphisms (but not group homomorphisms). Thus

the claim follows from the fact that π1(SLn(C)) = 1. There are various insightfulproofs of this fact. All proofs begin with the observation that SU(n) → SLn(C) isa homotopy equivalence by polar decomposition.

The standard proof of π1(SU(n)) = 1 uses the long exact homotopy sequence,see [19] §VII.8. In the lecture ”Topology I”, I gave a proof that uses only π1 andthe Seifert-Van Kampen theorem. Here is a sketch: we argue by induction on n,the case n = 2 serving as induction beginning (SU(2) ≅ S3, and this is simplyconnected). Consider the map p ∶ SU(n) → S2n−1 that takes a matrix A to Ae1.The map p is a fibre bundle with fibre SU(n− 1), as we will see later. Cover S2n−1

by the complements Ui, i = 0,1 of two different points. Then Ui is homeomorphicto R2n−1 and one can show that p is trivial over both subsets, in other words, thereare homeomorphisms p−1(Ui) ≅ Ui × SU(n − 1) over Ui. Now let Vi ∶= p−1(Ui).The sets Vi ≅ R2n−1 ×SU(n−1) are simply connected by induction hypothesis, andthe indersection V0 ∩ V1 ≅ (R2n−1 ∖ 0) × SU(n − 1) is connected. So, by Seifert-vanKampen, the union SU(n) = V0 ∪ V1 is simply connected.

Another interesting proof that uses a bit of the structure of the Lie group SU(n)is due to Hermann Weyl and can be found in Rossmann’s book [57].

There are stabilization maps st ∶ GLn(C) → GLn+1(C) which induce, by theprevious lemma, isomorphisms st∗ ∶ [S1,GLn(C)] → [S1; GLn+1(C)]. If Jn ∶[S1; GLn(C)]→ Z denotes the map [f]↦ ind(Tf), we get that

Jn+1 st∗ = Jn.This is nothing else that the observation that Tstf = Tf ⊕ id.

Corollary 1.7.3. Jn ∶ π1(GLn(C))→ Z; [f]↦ ind(Tf) is an isomorphism.

Let us switch the perspective a bit further. Suppose that f, g ∶ S1 → GLn(C) aretwo maps. We might now consider the direct sum f ⊕ g ∶ S1 → GL2n(C).Lemma 1.7.4. There are homotopies f ⊕ g ∼ fg ⊕ 1, f ⊕ g ∼ g ⊕ f .

Proof. We only give the first one, as the second is similar in spirit and equally easyto find. Look at

(f1)(cos(t) sin(t)

sin(t) − cos(t))(1g)(cos(t) sin(t)

sin(t) − cos(t))

For t = 0, we get f ⊕ g, for t = π/2, we get fg ⊕ 1.


We now get another, almost trivial, proof of the fact that ind(Tfg) = ind(Tf) +ind(Tg) (this took some work above!). Namely: ind(Tf⊕g) = ind(Tf) + ind(Tg) isobvious, and the above homotopy shows that ind(Tf⊕g) = ind(Tfg ⊕ 1) = ind(Tfg).

Passing to the colimit, we get a new description of the group structure onπ1(GL∞), namely the one given by direct sum, and it agrees with the old one.

We draw a lesson from these observations. By ”stabilizing”, we often have thepossibility to exchange the operation of composition by the operation of direct sum,which is often much easier to handle.

1.8. An example from ordinary differential equations. Let us discuss theone single case when the index of a differential operator can be computed by hand,namely the case of an ordinary differential operator on S1. What we do in effectis to prove the Atiyah-Singer index theorem for the manifold S1 by bare hands:each elliptic differential operator on S1 of order 1 has index zero (the final result isunfortunately quite boring). Via the usual map R/Z→ S1, t↦ e2πit, we can identify(vector-valued) functions on S1 with 1-periodic functions C∞(R;Cn)1. Now letA ∶ R → Matn,n(C) be a smooth, 1-periodic, matrix valued function. We considerthe linear differential operator

(1.8.1) D ∶ C∞(R;Cn)→ C∞(R;Cn); f ↦ f ′ +Af.

This is in fact an elliptic differential operator on S1, as we will learn soon.Because A is 1-periodic, D maps C∞(R,C)1 to itself, and we denote the restrictionby

(1.8.2) Dper =D ∶ C∞(R;Cn)1 → C∞(R;Cn)1.

Recall from Analysis II the solution theory of linear ODEs of order 1, forgettingfor the moment that A is assumed to be periodic. There exists a (unique) functionW ∶ R→ GLn(C) such that W (0) = 1 and W ′ = −AW , the fundamental solution. Ifv ∈ Cn, then f(t) =W (t)v is the unique solution to the initial value problem

Df = 0; f(0) = v.We also need to talk about inhomogeneous solutions, namely solutions f of the

ODE

(1.8.3) Df = u.

Let us try to solve the equation 1.8.3, first with the intial value f(0) = 0. Tofind the solution, we make the ansatz f(t) = W (t)c(t) for a yet to be determinedfunction c ∶ R→ Cn (with c(0) = 0). Applying the equation 1.8.3, we find that

c′ =W −1u or c(t) = ∫t

0W (s)−1u(s)ds.

The general solution to the initial value problem Df = u, f(0) = v is then givenby

(1.8.4) f(t) =W (t)v +W (t)∫t

0W (s)−1u(s)ds.

16 JOHANNES EBERT

We have proven so far that D ∶ C∞(Rn) → C∞(Rn) is surjective and has n-dimensional kernel. But we want to talk about periodic solutions. Assume that Ais 1-periodic and let W (t) be the fundamental solution. For all t ∈ R, the identity

(1.8.5) W (t + 1) =W (t)W (1)holds, as one proves by differentiating both sides of the equation and comparingthe values for t = 0. We consider the linear map ψ ∶ Cn → C∞(R;Cn), given by

v ↦W (t)vIf v is in the eigenspace ker(W (1) − 1), then W (t + 1)v =W (t)W (1)v =W (t)v,

and so W (t)v is a periodic function. But W (t)v is also a solution of the ODEDf = 0, and so ψ maps ker(W (1) − 1) to ker(Dper), and ψ is injective. But weknow that any periodic solution of Df = 0 can be written in the form W (t)v, andthis is periodic if and only if v is an eigenvector. Thus

ψ ∶ ker(W (1) − 1)→ kerDper is an isomorphism.

Now turn to the determination of the cokernel of the operator Dper. Let u be aperiodic function and suppose that there is a periodic solution of Df = u. Then

f(0) = f(1) =W (1)f(0) +W (1)∫1

0W (s)−1u(s)ds

or

W (1)−1(1 −W (1))f(0) = ∫1

0W (s)−1u(s)ds.

In other words, ∫1

0 W (s)−1u(s)ds lies in Im(W (1)−1(1−W (1))) = Im(1−W (1))W (1)−1) =Im(1 −W (1)). The previous manipulations can be read in the opposite direction,

which proves that a periodic solution Df = u exists iff Ju ∶= ∫1

0 W (s)−1u(s)ds ∈∫

10 W (s)−1u(s)ds ∈ Im(1 −W (1)). The linear map

J ∶ C∞(R,Cn)1 → Cn; Ju = ∫1

0W (s)−1u(s)ds

is surjective. To see this, take a function a ∈ C∞(R) with compact support in

(0,1) with ∫1

0 a(s)ds = 1. For v ∈ Cn, form u(s) ∶= a(s)W (s)v which has compactsupport and extend it to all of R by 1-periodicity. But

Ju = ∫1

0W (s)−1W (s)a(s)vds = v.

These arguments show that the image of Dper is the preimage J−1(Im(1 −W (1))) ⊂ C∞(R,C)1; and this preimage has the same codimension as Im(1 −W (1)) ⊂ Cn. But this codimension is, by the rank-nullity theorem, the same asdim(ker(W (1) − 1)), and hence

ind(Dper) = 0.

We go one step further. The vector space C∞(R,Cn)1 has an inner product

⟨f ; g⟩ ∶= ∫1

0 (f(t); g(t))dt, using the integral and the inner product on Cn. Now weconsider the adjoint operator to D:

D∗f(t) ∶= −f ′(t) +A(t)∗f(t).


Let V ∶ R → Matn,n(C) be the fundamental solution for D∗, i.e. V (0) = 1 andV ′ = A∗V .

Exercise 1.8.6. Prove:

(1) D∗ is indeed the adjoint of D in the sense that ⟨D∗f ; g⟩ = ⟨f ;Dg⟩ holds forall functions f, g (partial integration).

(2) V ∗W = 1 (differentiate!).(3) Im(W (1) − 1) = (ker(V (1) − 1)).(4) Conclude that u ∈ Im(D) if and only for all w ∈ ker(V (1)−1), the equation

∫1

0 (V (s)w,u(s))ds = 0 holds.(5) Prove that there is an orthogonal sum decomposition C∞(R;Cn)1 = Im(D)⊕

ker(D∗).

In two cases, there are explicit formulae for the solution operator. If n = 1, then

W (t) = exp(− ∫t

0 A(s)ds). The other easy case is when A(s) ≡ A is constant, inwhich case the fundamental solution is exp(At).

Exercise 1.8.7. Assume that n = 1. Prove that dim ker(D) = 1 if and only if

∫1

0 a(s)ds ∈ 2πi (in the other case, the kernel is trivial). Assume that n ≥ 1 and Ais constant. Show that dim(ker(D)) = ∑k∈Z Eig(A,2πk).

1.9. Literature and remarks. To aquire the neccessary background in functionalanalysis for the index theorem, you do not have to delve into the formidable treatise[58]; the nice book [38] contains all material. The proof of the basic properties ofcompact operators is taken from that source. The treatment of Fredholm operatorsis a ”best of” the (relevant sections) of the texts [38], [15], [36].

18 JOHANNES EBERT

2. Differential operators on manifolds and the de Rham complexrevisited

The index theorem will give a formula for the index of a general elliptic differ-erntial operator, but the main interest lies in special operators that are associatedwith any manifold or any manifold with some extra structure. The father of most ofthese natural operators is the exterior derivative, which we briefly recall. Then wemove on to the general definition of a differential operator on a manifold, introducethe symbol and the notion of ellipticity.

2.1. The de Rham operator. We are very brief in this section. The material isstandard and can be found in many textbooks, of which I most recommend [42] for afirst orientation and [68] for a more detailed exposition. Let V be an n-dimensionalreal vector space and let ΛpV ∗ be the space of alternating p-multilinear forms onV . There is the wedge product

ΛpV ∗ ⊗ΛqV ∗ → Λp+qV ∗;ω ⊗ η ↦ ω ∧ η,which turns Λ∗V ∗ into a graded commutative algebra, i.e.

ω ∧ (η ∧ ζ) = (ω ∧ η) ∧ ζ; ω ∧ η = (−1)∣ω∣∣η∣η ∧ ωwhere ∣ω∣ denotes the degree of ω. Moreover, for each v ∈ V , we have the insertion

operator ιv ∶ ΛpV ∗ → Λp−1V ∗, defined by (ιv1ω)(v2, . . . , vp) ∶= ω(v1, . . . , vp). It isan antiderivation, i.e.

ιv(ω ∧ η) = (ιvω) ∧ η + (−1)∣ω∣ω ∧ (ιvη).For ξ ∈ V ∗ = Λ1V ∗, let εξ(ω) ∶= ξ ∧ ω. Both structures, the exterior product and

the insertion operator, are intertwined by the easily verified identity

(2.1.1) εξιv + ιvεξ = ξ(v)(left hand-side is the operator that multiplies by ξ(v)). Let Mn be a smooth

manifold. A vector field on M is a section of TM →M , and V(M) ∶= Γ(M ;TM)denotes the space of all vector fields on M . Tangent vectors to a manifold havea schizophrenic nature (as derivatives of curves and directional derivatives). Thismeans that V(M) has the alternative expression as the set of all linear map X ∶C∞(M)→ C∞(M) such that X(fg) = (Xf)g + f(Xg) holds for all functions f , g.The commutator

[X,Y ] ∶=XY − Y Xof two vector fields is again a vector field. The commutator is also called Lie

bracket and it turns V(M) into a Lie algebra. Let Ap(M) be the space of allsmooth p-forms on M . One can interprete Ap(M) as the space of smooth sectionsof the bundle ΛpT ∗M →M of exterior forms on the tangent bundle and thereforethe linear algebraic structure on the exterior algebra (wedge product and insertionoperator) carries over to forms on manifolds.

The most important structure is the exterior derivative, a sequence of linearmaps d ∶ Ap(M)→ Ap+1(M), uniquely characterized by the properties

On A0(M) = C∞(M), d is the total differential.

d(ω ∧ η) = (dω) ∧ η + (−1)∣ω∣ω ∧ dη (Leibniz rule) d2 = 0.


A smooth map f ∶M → N induces a map f∗ ∶ A(N) → A(M), compatible withd and ∧. The quotient space

HpdR(M) ∶= ker(d ∶ Ap(M)→ Ap+1(M))

Im(d ∶ Ap−1(M)→ Ap(M))is called de Rham cohomology of M . The de Rham cohomology has a deep

topological meaning, which shall not bother us right now. Instead, we look at theoperator d in more detail. Let x ∶ M ⊃ U → Rn be a local coordinate system onM . For a subset I = i1, . . . , ip ⊂ n, ∣I ∣ = p, let dxI ∶= dxi1 ∧ . . . ∧ dxip . In thesecoordinates, we can write each p-form (locally) as

ω = ∑I⊂n;∣I ∣=p

aIdxI

for smooth functions aI . The exterior derivative is then given by

dω = ∑I⊂n;∣I ∣=p

n

∑j=1

∂aI∂xj

dxj ∧ dxI .

This formula shows that d is a linear partial differential operator of order 1, anotion with which we will have to familiarize us next. We can combine the exteriorderivative with the insertion operator to get the Lie derivative

LX(ω) ∶= d(ιXω) + ιXdω.The Lie derivative takes p-forms to p-forms, and has the following properties

Proposition 2.1.2.

(1) LX commutes with ιX and d.(2) LX(ω ∧ η) = (LXω) ∧ η + ω ∧LXη (no sign).(3) if f ∈ A0(M), then LXf =Xf .

2.2. Differential operators in general.

Notation 2.2.1. For a multiindex α = (α1, . . . , αn) ∈ Nn, we let ∣α∣ ∶= ∑i αi. For

x ∈ Rn, we let xα ∶= ∏ni=1 x

αii and ∂ ∣α∣

∂xα∶= ∂α1

∂x1. . . ∂

αn

∂xn. Moreover, Dα ∶= (−i)∣α∣ ∂

∣α∣

∂xα.

Furthermore, α! ∶=∏ni=1 αi!.

If E →M is a vector bundle over K = R or C, we denote by Γ(M,E) the vectorspace of smooth sections of E.

Definition 2.2.2. Let M be a smooth manifold and Ei →M be two smooth vectorbundles. A differerential operator P ∶ Γ(M,E0) → Γ(M ;E1) of order k is a linearmap which satisfies the following properties:

(1) P is local in the sense that if s ∈ Γ(M,E0) vanishes on the open subsetU ⊂M , then so does Ps.

(2) If x ∶ U → Rn is a chart and φi ∶ Ei∣U → U ×Kpi a trivialization, then thelocalized operator φ1 P (φ0)−1 can be written as

(φ1 P (φ0)−1)(f)(y) = ∑∣α∣≤k

Aα(y) ∂α

∂xαf(y)

for each f ∈ C∞(U,Kp0), where Aα ∶ U →Matp1,p0(K) is a smooth function.

20 JOHANNES EBERT

Examples 2.2.3. Composition with a vector bundle homomorphism induces anoperator of order 0. The exterior derivative is an operator of order 1.

There is a coordinate-free description of differential operators, which is some-times useful.

Theorem 2.2.4. Let P ∶ Γ(M,E0)→ Γ(M,E1) be linear. Then

(1) P is a differential operator of order 0 if and only if the commutator withthe multiplication by any function f ∈ C∞(M) is zero; [P, f] = 0.

(2) P is a differential operator of order k if and only if the commutator [P, f]is an operator of order k − 1, for each f ∈ C∞(M).

Proof. In both parts, the ”only if” direction is easy and we turn to the ”if” direction.Part (1). First we prove that P is local. If s ≡ 0 near x, there is a function f withf ≡ 1 near x and fs = 0. Then

Ps(x) = fPs(x) = P (fs)(x) = 0

and hence P is local. Thus we can compute in local coordinates. Let s1, . . . , sp0be a local basis of E0 and ai ∈ C∞. We find

P (∑i

aisi)(x) =∑i

ai(x)Psi(x).

Psi is a section of E1 and we can write it as a linear combination of a given localbasis of E1 with smooth coefficients. This gives the desired presentation of P .

Part (2). Assume that [P, f] is an operator of order k − 1, for each f ∈ C∞(M).We first prove that P is local. As above, assume that f ≡ 1 and s ≡ 0 near x. Then

Ps(x) = fPs(x) = [f,P ]s(x) + P (fs)(x) = [f,P ]s(x).By induction hypothesis, [f,P ] is local and so [f,P ]s(x) = 0. Let x0 ∈ M and

x a local chart such that x(x0) = 0. Next, we recall a lemma that was crucial inproving that the tangent space of a manifold, defined using derivations, was ann-dimensional vector space. Let 0 ∈ U ⊂ Rn be convex and f ∶ U → R be smooth.Then

f(x) = f(0) + ∫1

0

∂

∂tf(tx)dt = f(0) +

n

∑i=1∫

1

0xi∂f

∂xi(tx)dt =∶ f(0) +

n

∑i=1

xigi(x)

with gi smooth and gi(0) = ∂∂xi

f(0). Iteratively, we find that there is a unique

polynomial p(x) of degree k and smooth functions gα, ∣α∣ = k + 1 with gα(0) =∂ ∣α∣

∂xαf(0) such that

(2.2.5) f(x) = p(x) + ∑∣α∣=k+1

xαgα(x).

Moreover, if [P, f] has order k − 1 for each f , we find that for all functionsf0, . . . , fk with fi(0) = 0, we have

P (f0 . . . fks)(0) = 0

for each section s, because


P (f0 . . . fks)(0) = [P, f0](f1 . . . fks)(0) + f0(0)P (f1 . . . fks)(0) = 0

by induction hypothesis. If s1, . . . , sp0 is a local basis and ai smooth and x(x0) =0, then

P (∑i

aisi)(x0) =∑i

P (aisi)(x0) =∑i

P ((pi + ∑∣α∣=k+1

xαgi,α)si)(x0)

But as [P, f] has order k − 1, we find P ((pi + ∑∣α∣=k+1 xαgi,α)si)(x0) = 0 and

hence P (∑i aisi)(x0) = ∑i P (pisi)(x0). But pi(x) = ∑∣α∣≤k1α!

∂ ∣α∣

∂xαai(0)xα and thus

P (aisi)(x0) = ∑∣α∣≤k1α!

∂α

∂xαai(0)P (xαs)(x0). Rearranging all terms gives the de-

sired presentation.

Example 2.2.6. Let E0 = E1 = R ∶= M × R be the trivial bundle. Let X be avector field, i.e. a derivation, in other words a map X ∶ C∞(M) → C∞(M) suchthat X(fg) = (Xf)g + f(Xg) for all f, g ∈ C∞(M). The commutator [X,f] is theoperator

[X,f]g = (Xf)gand so it is an operator of order 0. Thus, X is a differential operator of order 1.

Example 2.2.7. According to the list of axioms for the exterior derivative, we cancompute the commutator [d, f] as

[d, f]ω = d(fω) − fdω = df ∧ ωand so d is a differential operator of order 1.

Example 2.2.8. If E → M is an arbitrary vector bundle, a connection on E isa linear map ∇ ∶ Γ(M,E) → Γ(M,T ∗M ⊗ E) such that for all s ∈ Γ(M,E) andf ∈ C∞(M), we have ∇(fs) = df ⊗ s + f∇s, which means that

[∇, f]s = df ⊗ swhich is why a connection is a differential operator of order 1, characterized by

the condition [∇, f] = df ⊗ .

Definition 2.2.9. Let M be a manifold and E0,E1 be two vector bundles. Wedenote by Diffk(E0,E1) the set of all differential operators of order k.

Let us note some obvious properties.

Lemma 2.2.10.

(1) Diffk(E0,E1) is a vector space.

(2) Diffk(E0,E1) ⊂ Diffk+1(E0,E1).

(3) If P ∈ Diffk(E0,E1) and Q ∈ Diffm(E1,E2), then Q P ∈ Diffk+m(E0,E2).

We remark that the order of P is not really a well-defined number. Some ofthe most important information about a differential operator can be encoded inthe terms of highest order, the symbol. From now on, we assume that the vectorbundles Ei are over C.

22 JOHANNES EBERT

Definition 2.2.11. Let P ∈ Diffk(E0,E1). Let y ∈ M , ξ ∈ T ∗yM and e ∈ (E0)y.Pick f ∈ C∞(M ;R) with f(y) = 0 and dyf = ξ, and pick s ∈ Γ(M,E0) with s(y) = e.We define the symbol of P to be

smbk(P )(y, ξ)(e) ∶= ik

k!P (fks)(y) ∈ (E1)y.

Lemma 2.2.12. The expression smbk(P )(y, ξ)(e) only depends on y, ξ and e (as-suming that P and k are fixed).

Proof. For the purpose of this proof, we compute in local coordinates. Pick a localchart with x(y) = 0. We can assume that the bundles E0 and E1 are trivial overthe domain of the chart x. We write ξ = ∑i ξidxi, ξi ∈ R and compute

ik

k!P (fks)(x) = i

k

k!∑

∣α∣≤kAα(x) ∂

∣α∣

∂xα(fks)(x) = i

k

k!∑

∣α∣≤k∑

β+γ=α

α!

β!γ!Aα(x) ∂

∣β∣

∂xβ(fk)(x) ∂

∣γ∣

∂xγ(s)(x).

Since the derivative ∂ ∣β∣

∂xβfk(x) is zero for ∣β∣ < k (the argument from the proof of

Theorem 2.2.4), the sum equals

ik

k!∑

∣α∣=kAα(x) ∂

∣α∣

∂xα(fk)(x)s(x).

But ∂ ∣α∣

∂xα(fk)(x) = k!ξα, and we conclude that

(2.2.13) smbk(P )(y, ξ)(e) = ik ∑∣α∣=k

Aα(x)ξαe.

This equation can be read in two directions: it shows that the left-hand-sidedoes not depend on the concrete choice of f and s, and that the right-hand-sidehas a coordinate-invariant meaning.

Remark 2.2.14. There are in principle two approaches to calculus on manifolds(the coordinate-free one and the one using coordinates). As a general rule, thecoordinate-free approach is more modern and preferred by most pure mathemati-cians (for good reasons). The above proof shows that the combination of both canbe a useful argument, and that one should not stick ideologically to the coordinate-free approach.

Exercise 2.2.15. If you think that using local coordinates is a stupid thing to do,try to give a proof of Lemma 2.2.12 and Proposition 2.2.18 avoiding choices ofcoordinates (this is possible).

Now we give a more invariant interpretation of the symbol. Let V be a finite-dimensional real vector space (think about TxM). By the symbol SymkV , wedenote the vector space of degree k homogeneous polynomial functions V ∗ → R.Given a manifold M , we can form the vector bundle SymkTM → M , whose fibreover x is precisely SymkTxM . What does the symbol do? It assigns to any givenξ ∈ T ∗xM a linear map smbk(P )(x, ξ) ∶ (E0)x → (E1)x, and ξ ↦ smbk(P )(x, ξ)is a polynomial function T ∗xM → Hom((E0)x, (E1)x). Moreover, this polynomialfunction is homogeneous of degree k. To see this, simply look at the right-hand sideof 2.2.13. We might now form the vector bundle SymkTM ⊗ Hom(E0,E1) → M ,


and the symbol smbk(P ) defines a section of this vector bundle: at a point x ∈M ,the value of this section is the polynomial function ξ ↦ smbk(P )(x, ξ). Anotherlook at 2.2.13 proves that this is a smooth section.

Definition 2.2.16. Let Smblk(E0,E1) be the space of smooth sections of SymkTM⊗Hom(E0,E1)→M . The symbol of an order k operator is a well-defined element of

Smblk(E0,E1) and we have produced a map smbk ∶ Diffk(E0,E1)→ Smblk(E0,E1).

Example 2.2.17. It is worth to work out the meaning of all this in the case whenM = U ⊂ Rn is an open subset, both vector bundles are trivialized and P is adiffererential operator of order k. We write

Pu(x) = ∑∣α∣≤k

Aα(x)Dαu(x)

(note that we used Dα here, instead of ∂ ∣α∣

∂xαas above). The symbol is now given

(using 2.2.13) by the formula

smbk(P )(x, ξ) = ∑∣α∣=k

Aα(x)ξα.

We can represent a differerential operator P on U ⊂ Rn by a function p(x, ξ)which is smooth in x and polynomial (of degree k) in ξ. The operator P is givenby p(x,D) (replace ξi by Di). Some caution is necessary because Di does notcommute with multiplication by smooth functions. One sometimes calles the poly-nomial p(x, ξ) the complete symbol, and the leading part the principal symbol. Ona manifold, the complete symbol does not have an easily identified meaning, onlythe principal symbol (which we call ”symbol”) does.

It is obvious that smbk is a linear map (if you are unsure, look at the formula2.2.13). If Ei, i = 0,1,2, are three vector bundles and k, l two natural numbers,there is a composition map

Smblk(E0,E1) × Smbll(E1,E2)→ Smblk+l(E0,E2).This is defined by means of linear algebra: If V is a finite-dimensional real vector

space and Ei, i = 0,1,2, complex vector spaces, then a bilinear map

(SymkV ⊗Hom(E0,E1)) × (SymlV ⊗Hom(E1,E2))→ (Symk+lV ⊗Hom(E0,E2))is given by

(p⊗ a, q ⊗ b)↦ pq ⊗ (b a).More concretely, we compose an order l homogeneous polynomial p on T ∗xM

with values in Hom((E1)x, (E2)x) with an order k homogeneous polynomial q withvalues in Hom((E0)x, (E1)x). This is done by ξ ↦ (p q)(ξ) ∶= p(ξ) q(ξ).

Proposition 2.2.18. Let P ∈ Diffk(E0,E1) and Q ∈ Diff l(E1,E2). Then smbk+l(QP ) = smbl(Q)smbk(P ).

Proof. In concrete terms, this means that for each cotangent vector ξ ∈ T ∗xM , wehave smbk+l(Q P )(ξ) = smbl(Q)(ξ) smbk(P )(ξ), which is easy to see usingformula 2.2.13.

24 JOHANNES EBERT

Proposition 2.2.19. The sequence

0→ Diffk−1(E0,E1)→ Diffk(E0,E1)→ Smblk(E0,E1)→ 0

is exact.

Proof. Everything is easy (for example in local coordinates) except exactness at theright. We have to show: if p ∈ Smblk(E0,E1), then there is a differential operator Pwith symbol p. Locally (in local coordinates) the problem is easy to solve, becauseif p is given by

∑∣α∣=k

Aα(x)ξα

with some matrix-valued functions Aα, the operator

∑∣α∣=k

Aα(x)Dα

has the required symbol. To glue the local solutions together, one uses a partitionof unity.

All this becomes more transparent if we consider operators of order 1. Assumethat P has order 1, x ∈M , f ∈ C∞(M) and s ∈ Γ(M,E0). Then compute

[P, f]s(x) = [P, f − f(x)]s(x) + [P, f(x)]s(x) = [P, f − f(x)]s(x).(note that f(x) denotes the constant function, and note that P therefore commuteswith f(x)). Moreover,

[P, f−f(x)]s(x) = P ((f−f(x)(s))(x)−(f−f(x))(x)Ps(x) = P ((f−f(x))(s))(x) = −ismb1(P )(df)s(x).Thus:

Lemma 2.2.20. If P is an operator of order 1, the symbol can be computed assmb1(P )(df)s ∶= i[P, f]s.

Now, by definition, Smbl1(E0,E1) is the space of sections in the vector bundle

Sym1TM ⊗Hom(E0,E1) = TM ⊗Hom(E0,E1) = Hom(T ∗M ⊗E0;E1)and in this description, the symbol of P becomes smb1(P )(df)s ∶= i[P, f]s. Here

we used a section and the derivative of a function as a variable, but Lemma 2.2.12proves that this is really a well-defined section of vector bundles.

We now come to one of the central definitions of this lecture course.

Definition 2.2.21. Let M be a smooth manifold, E0,E1 →M two complex vectorbundles and P ∈ Diffk(E0,E1), x ∈ M . Then we say that P is elliptic at x if foreach ξ ∈ T ∗xM , ξ ≠ 0, the homomorphism smbk(P )(ξ) ∶ (E0)x → (E1)x is invertible.We say that P is elliptic if it is elliptic at each point of M .

Example 2.2.22. Let M = R and A,B ∶ R → Matp,p(C). Consider the operatorP ∶ C∞(R;Cp)→ C∞(R;Cp) given by

Pf ∶= Bf ′ +Af = B ∂

∂xf +Af.


Let us compute the symbol. A typical cotangent vector vector is (x, ξdx), x ∈R(= M), ξ ∈ R. In this local coordinate, we find that the symbol is given bysmb1(P )(ξdx) = iξB(x) ∈ Matp,p(C). Thus we find that P is elliptic at x ∈ R ifand only if B(x) is invertible.

Example 2.2.23. From basic complex analysis, one knows the Wirtinger or Cauchy-Riemann operators on C∞(U ;C), where U ⊂ C is open. They are defined by

∂

∂z∶= 1

2( ∂∂x

− i ∂∂y

); ∂

∂z∶= 1

2( ∂∂x

+ i ∂∂y

)

(x and y are the real coordinates z = x+ iy). Let adx+ bdy be a (real) cotangentvector to U . We find that the symbol of ∂

∂zis given by

smb1(∂

∂z)(adx + bdy) = i1

2(a + ib).

(substitute the jth cotangent coordinate for the partial derivative Di). We canidentify T ∗C with C in the canonical way, this corresponds to adx + bdy ↦ a + ib.Call the resulting coordinate ζ (this is a complex-valued linear form on T ∗U). Thusthe symbol of ∂

∂zis multiplication by i

2ζ.

Similarly, one finds that

smb1(∂

∂z)(adx + bdy) = i1

2(a − ib) = i

2ζ.

Both operators are elliptic. The holomorphic functions on U are precisely thesolutions of the PDE ∂

∂zf = 0, and this remark shows that complex analysis in

one variable is a special case of elliptic operator theory (in higher dimensions, thesituation is much more subtle).

In the theory of Riemann surfaces, there is an important operator combiningthe two Wirtinger operators. Namely, let µ ∈ C∞(U) be a smooth function andconsider P = ∂

∂z+ µ ∂

∂z. The symbol is

smb1(P )(ζ) = i

2(ζ + µζ).

For which µ is this operator elliptic? To get to the important point, write V =T ∗xU and note that ζ ∶ V → C is a (real-linear) isomorphism. In this reformulation,the problem becomes to find under which conditions on a complex number µ = µ(x),the equation ζ + µζ has no nontrivial solutions for ζ ∈ C. Observe that ∣ζ ∣ = ∣ζ ∣ andtherefore, if ζ +µζ = 0 for ζ ≠ 0, we must have ∣µ∣ = 1. Thus if P is not elliptic at x,then ∣µ∣ = 1. Vice versa, if ∣µ(x)∣ = 1, then P is not elliptic at x.

Thus: the operator P is elliptic on U if ∣µ∣ ≠ 1 on U . The relevant case is when∣µ∣ < 1. The operator P is relevant for the problem of integrability of almost-complexstructures.

Example 2.2.24. Let us compute the symbol of the exterior derivative d ∶ Ap(M)→Ap+1(M). Let f ∈ C∞(M). Then

smb1(d)(df)ω = i(d(fω) − fdω) = i(df ∧ ω).Thus the symbol of d, viewed as a bundle map T ∗M ⊗ ΛpT ∗M → Λp+1T ∗M , is

given by (ξ, ω)↦ iξ ∧ ω.

26 JOHANNES EBERT

Thus, the exterior derivative is not elliptic unless dimM = 1. However, it isrelatively close to being elliptic.

Lemma 2.2.25. Let V be a finite dimensional real vector space of dimension n.For ξ ∈ V ∗, denote by εξ ∶ ΛpV ∗ → Λp+1V ∗ the map εξω ∶= ξ ∧ ω. Then if ξ ≠ 0, thesequence

0→ Λ0V ∗ εξ→ Λ1V ∗ εξ→ . . .εξ→ ΛnV ∗ → 0

is exact.

Proof. For v ∈ V , we get the map ιv ∶ ΛpV ∗ → Λp−1V ∗ which inserts v as the firstargument. It satisfies ιv(ω ∧ η) = (ιvω) ∧ η + (−1)∣ω∣ω ∧ ιvη. By 2.1.1

(ιvεξ + εξιv)ω = (ξ(v))ω.Now pick v such that ξ(v) = 1. If ξ ∧ ω = 0, we find that

ω = ξ(v)ω = (ιvεξ + εξιv)ω = εξ(ιvω)and this proves the exactness.

This lemma proves that the de Rham complex fits into the following definition.

Definition 2.2.26. Let M be a smooth manifold. An elliptic complex of length nis a sequence

0→ Γ(M,E0)P1→ Γ(M,E1)

P2→ . . .Pn→ Γ(M,En)→ 0

of differential operators of order 1 between complex (or real) vector bundles suchthat

(1) Pi Pi−1 = 0 and(2) for each nonzero cotangent vector ξ ∈ T ∗xM , the sequence

0→ (E0)xsmb1(P1)(ξ)Ð→ (E1)x

smb1(P2)(ξ)Ð→ . . .smb1(Pn)(ξ)Ð→ (En)x → 0

is exact.

We write the elliptic complex as (E∗, P ).Remark 2.2.27. An elliptic complex of length one is the same as an elliptic opera-tor on M . The length and the dimension of M are completely unrelated in general.One can formulate a more general definition where the operators Pi have order > 1,but for applications this is irrelevant. It is important that all operators have thesame order.

Out of an elliptic complex, we can extract an elliptic operator, but that requiresRiemann metrics on M and hermitian bundle metrics on the bundles Ei.

2.3. The formal adjoint.

Assumptions 2.3.1. The manifold M comes equipped with a Riemann metric,and the complex vector bundles are equipped with hermitian bundle metrics.

We first recall how functions can be integrated on a Riemannian manifold. Firstassume that M is an oriented Riemann manifold of dimension n. There is a uniquen-form vol ∈ An(M), characterized by the property that if (e1, . . . , en) is an orientedorthonormal basis of T ∗xM , for some x ∈M , then vol(e1, . . . , en) = 1. The form volis called the volume form.


Definition 2.3.2. Let f ∈ C∞c (M) be a function with compact support. We define

∫Mf(x)dx ∶= ∫

Mfvol.

This notion of integral uses the orientation on M , but it does use the orientationtwice: to define the volume form and to define the integral. Both dependencescancel out: assume that T ∶M → N is an orientation-reversing isometry. Then

∫Mf(Tx)dx = ∫

M(T ∗f)volM

1= −∫MT ∗(fvolN) 2= ∫

NfvolN = ∫

Nf(x)dx.

The equation 1 holds since T ∗volN = −volM because T reverses orientation; theequation 2 holds because the integral of forms depends on the orientation. Anotherway to express this is by saying that f ≥ 0, then ∫M f ≥ 0. We might extendthe definition of the integral to nonoriented Riemann manifolds as follows. Letπ ∶ M →M be the orientation cover. The manifold M has a canonical orientation.The Riemann metric on M gets pulled back to M , and the unique nontrivial Decktransformation T of the covering M →M becomes an orientation-reversing isometryof M . Now we define

∫Mf(x)dx ∶= 1

2∫M

(f π)(x)dx.

The factor 1/2 guarantees that the new integral coincides with the old one onoriented manifolds. It is easy to see ∫M f(x)dx ≥ 0 for f ≥ 0, and also that if f ≥ 0has integral 0, then f = 0.

Note that this procedure does not give a sensible procedure to integrate n-formson a nonoriented manifold. If ω ∈ An(M), then ∫M π∗ω = − ∫M T ∗π∗ω = − ∫M π∗ω,and so the integral is zero.

Definition 2.3.3. Let E → M be a hermitian vector bundle on a Riemannianmanifold. Let s, t ∈ Γc(M ;E). By

⟨s, t⟩ ∶= ∫M

(s(x), t(x))dx,

we define an inner product on the space of compactly supported sections of E,which therefore becomes a pre-Hilbert space. We let L2(M ;E) be the Hilbertspace obtained by completing Γc(M ;E) with respect to the norm given by thisscalar product.

Remark 2.3.4. We can make contact to measure theory as follows. The integralis a functional C0

c (M)→ C and has the property that ∫M f(x)dx ≥ 0 if f ≥ 0. Thus,by the Riesz representation theorem, [59], Thm 2.14, there is a unique measureon the σ-algebra of Borel sets that gives this functional by integration. The usualtheorems from Analysis III show that we can view the elements of the Hilbert spaceL2(M,E) as measurable sections of the vector bundle E.

Definition 2.3.5. Let M be riemannian and E0,E1 → M be hermitian, and P ∈Diffk(E0,E1). A formal adjoint P ∗ is a differential operator P ∗ ∶ Γ(M,E1) →Γ(M ;E0) such that for all compactly supported sections s, t, we have ⟨s,P ∗t⟩ =⟨Ps, t⟩. A differerential operator P is formally selfadjoint if E0 = E1 and if P ∗ = P .

28 JOHANNES EBERT

Theorem 2.3.6.

(1) The adjoint, if it exists, satisfies (PQ)∗ = Q∗P ∗, (Q + P )∗ = Q∗ + P ∗,(aP )∗ = aP ∗.

(2) The adjoint is uniquely determined.

(3) Each differential operator P ∈ Diffk(E0,E1) has an adjoint in Diffk(E1,E0).(4) The symbol of the adjoint can be computed pointwise: smbk(P ∗)(ξ) =

(smbk(P )(ξ))∗, where we used the adjoint of a vector bundle homomor-phism.

Before we give the proof, let us remark that ⟨s, t⟩ is defined if only one of thesections s and t has compact support. Moreover, if P ∗ is an adjoint of P , then⟨Ps, t⟩ = ⟨s,P ∗t⟩ holds if only one section has compact support.

Proof. The first part is trivial. If P ∗, P ′ are two adjoints, one has

⟨P ∗s, t⟩ = ⟨s,P t⟩ = ⟨P ′s, t⟩or

⟨(P ∗ − P ′)s, t⟩ = 0

for all sections s and t, which proves that P ∗s = P ′s for all s.For the existence of adjoints, we first prove that forming adjoints is a local proce-

dure, despite its appearance. So let U,V ⊂M be open, and let (P ∣U) and (P ∣V ) bethe restrictions of P to U,V . Assume that there exist adjoints (P ∣U)∗ and (P ∣V )∗.We now assert that (P ∣U)∗∣U∩V = (P ∣V )∗∣U∩V , in other words, the restrictions ofthe adjoints to the intersection agree. Let s, t be sections supported in U ∩V . Then

⟨(P ∣U)∗s, t⟩ = ⟨s,P t⟩ = ⟨(P ∣V )∗s, t⟩which proves the assertion. Assume that (Ui) is an open covering of M and P ∗

i

an adjoint of P ∣Ui . These operators fit together to a (differential) operator Q. Weclaim that Q is an adjoint of P . Let s, t be compactly supported sections and let(µi) be a partition of unity subordinate to (Ui). Then

⟨Ps, t⟩ =∑i,j

⟨Pµis, µjt⟩ =∑i,j

⟨µis,Qµjt⟩ = ⟨s,Qt⟩

(note that the sums are finite).To find the adjoints of the localized operators, we can assume that M = Rn, but

we cannot assume that the metric on M is the standard metric. Moreover, eachcomplex vector bundle has local trivializations which are isometric, i.e. preservethe inner product. To see this, begin with any local trivialization (which yields alocal basis) and apply the Gram-Schmidt process to it. The Gram-Schmidt processproduces a smooth orthonormal local basis. So, we assume that P is a differentialoperator on trivial vector bundles, with the standard bundle metric, over Rn, withsome metric.

We can write P = ∑∣α∣≤kAα(x)Dα, with some smooth matrix-valued functions

Aα. One way to argue from here is that Di has an adjoint. Another way is thatP is a sum of operators, each of which is given by matrix-multiplication (order 0operator) and a composition of vector fields. Thus it is enough to prove that order 0operators have adjoints (which is clear: take the pointwise adjoint) and that vectorfields X have adjoints.


Let f, g be two compactly supported functions. By Stokes theorem, we have

0 = ∫Md(ιX(fgvol)) = ∫

MLX(fgvol) = ∫

M(Xf)(gvol)+∫

Mf(Xg)vol+∫

MfgLXvol.

We define the divergence div(X) of the vector field X as the unique function

such that div(X)volM = LXvolM . This, together with Xf =Xf , shows that

⟨Xf, g⟩ = ∫MXfgvol = −∫

Mf(Xg)vol − ∫

MfgLXvol = −⟨f,Xg⟩ − ⟨f,div(X)g⟩;

in other words, that −X − div(X) is an adjoint of X.The formula for the symbols follows because it is true for order 0 operators and

for vector fields:

smb1(X∗) = −smb1(X).Since the symbol smb1(X) is skew-adjoint (since it is purely imaginary), the

formula for the symbol of an adjoint follows.

Let (E∗, P ) be an elliptic complex over the Riemannian manifold M . We assumethat each bundle Ei →M has a hermitian bundle metric. We get a bundle metric onthe direct sum⊕iEi, by requiring that the vector bundles Ei and Ej are orthogonalif i ≠ j. By taking the direct sum of the operators Pi, we get an operator P ∶⊕i Γ(M ;E)→⊕i Γ(M ;Ei) and the adjoint P ∗. We write Eev ∶=⊕iE2i and Eodd ∶=⊕iE2i+1. Note that P maps Γ(M,Eev) into Γ(M,Eodd) and vice versa. The sameapplies to the adjoint.

Proposition 2.3.7. Let (E∗, P ) be an elliptic complex and E = ⊕i≥0Ei. Thenthe operator P + P ∗ on Γ(M ;E) is elliptic (and formally self-adjoint). Hence therestricted operators P + P ∗ ∶ Γ(M,Eev) → Γ(M,Eodd) and P + P ∗ ∶ Γ(M,Eodd) →Γ(M,Eev) are elliptic.

This follows immediately, using Theorem 2.3.6 (4), from the following linearalgebraic lemma.

Lemma 2.3.8. Let 0 → V0f1→ V1

f2→ . . . Vn → 0 be a cochain complex of finitedimensional hermitian vector spaces. Then the following are equivalent:

(1) The complex is exact.(2) The linear map f + f∗ ∶ V∗ → V∗ is an isomorphism.

Proof. 2⇒ 1: The maps f∗ define a chain homotopy, from 0 to f∗f+ff∗ = (f+f∗)2.If f + f∗ is an isomorphism, then so is (f + f∗)2 and this isomorphism is chainhomotopic to 0. Thus the zero map induces an isomorphism on cohomology andthe complex is exact.

1⇒ 2: Let (f + f∗)x = 0. Then (ff∗ + f∗f)x = 0 and therefore

0 = ⟨ff∗x;x⟩ + ⟨f∗fx;x⟩ = ⟨f∗x; f∗x⟩ + ⟨fx; fx⟩,which is why fx = f∗x = 0. Since the complex is exact, there is y with fy = x, andy satisfies f∗fy = 0. Thus

0 = ⟨f∗fy; y⟩ = ⟨fy; fy⟩

30 JOHANNES EBERT

which implies x = fy = 0. Therefore f + f∗ is injective, and surjective by thefiniteness assumption.

2.4. Literature. The material covered in this section is largely standard material.The basic elements of calculus on manifolds are developped in great detail in thebook [68] (and in many other sources). The treatment of differential operators ingeneral is taken from [15].


3. Analysis of elliptic operators

We now have to delve into some nontrivial analysis. The goal is to state preciselyand prove the following two theorems.

Theorem 3.0.1. (Local regularity) If P is an elliptic differential operator, f asmooth section and let Pu = f . Then u is smooth.

At the moment, we only know what P should do to a smooth section, and soas stated, the Theorem is quite tautological. One way to phrase the theorem in anontrivial way is to assume that u is only Ck (in which case Pu still makes sense,if k is the order of P ). One example for this result is the well-known result fromcomplex analysis that a holomorphic function (which is assume to be C1) has tobe C∞.

In fact, this is not the intended precise formulation of the local regularity the-orem. What we will do is to introduce Hilbert spaces of ”weakly differentiablefunctions”, the Sobolev spaces, on which we can give a meaning to the equationPu = f , when u and f are Sobolev functions. The local regularity theorem willthen say that any solution in the Hilbert space sense is actually a smooth section.

The second main theorem of this section is

Theorem 3.0.2. If P is an elliptic differential operator on a closed manifold, thenP ∶ Γ(M,E0)→ Γ(M,E1) is a Fredholm operator.

Even if the theorem as stated is perfectly true, what we really prove is that Pinduces a map of certain Sobolev spaces, and that this map is a Fredholm operator(in the Hilbert space sense).

3.1. Preliminaries: Convolution and Fourier transformation.

Convention 3.1.1. Let dx be the normalized Lebegue measure on Rn, such thatthe unit cube [0,1]n has measure (2π)−n/2. The effect is that the Gaussian integralis normalized

∫Rne−x

2/2dx = 1

and that a lot of factors of the form (2π)±n/2 disappear.

For a complex valued function f on Rn, we have the Lebesgue norms

∥f∥Lp ∶= (∫Rn

∣f(x)∣pdx)1/p

whenever this makes sense. If f is a Ck function, we let

∥f∥Ck ∶= ∑∣α∣≤k

supx∈Rn

∣Dαf(x)∣

which is only meaningful if all derivatives up to order k are bounded.

Definition 3.1.2. Let f, g ∈ L1(Rn). The convolution f ∗ g is the function

f ∗ g(x) ∶= ∫Rnf(y)g(x − y)dy = ∫

Rnf(x − z)g(z)dz = g ∗ f(x).

The following important properties will be used many times. For the proof, see[46], p. 223 ff.

32 JOHANNES EBERT

Proposition 3.1.3.

(1) f ∗ g ∈ L1.(2) The convolution is associative, commutative and bilinear.(3) If f ∈ L1, g ∈ Lp, then f ∗ g ∈ Lp and ∥f ∗ g∥Lp ≤ ∥f∥L1∥g∥Lp (1 ≤ p ≤∞).(4) If f is smooth with compact support and g ∈ L1, then f ∗ g is smooth and

Dα(f ∗ g) = (Dαf) ∗ g.

Proposition 3.1.4. Let φ ∈ C∞c (Rn) be a function with φ ≥ 0 and ∫ φ(x)dx = 1.

Let, for t > 0, φt(x) = 1tnφ(x

t). Let f ∈ Lp. Then φt ∗ f converges, for t → 0, to f ,

in the Lp-norm. (p <∞).If f is a bounded continuous function, then φt ∗ f → f uniformly on all compact

subsets.

Corollary 3.1.5. The space C∞c (Rn) is dense in Lp(Rn), p <∞.

Proof. Let an be sequence of bump functions with increasing support. Then, foreach f ∈ Lp, anf is in Lp, and anf → f in Lp. Thus it is enough to approximatea function with compact support by smooth functions. Let f be such a function.The function φt ∗ f has compact support, and for t→ 0, it converges to f .

Definition 3.1.6. The space S(Rn) of Schwartz functions is the space of all smoothfunctions f such that for all multiindices α,β, the function xαDβf is bounded.

Examples: functions with compact support, e−x2

.

Definition 3.1.7. Let f ∈ S(Rn). The Fourier transform of f is defined as

f(ξ) ∶= ∫Rne−ixξf(x)dx.

We remark that one should really view ξ as a point in the dual space of Rn.Also, the functions e−ixξ are the characters of the group Rn, i.e. the continuoushomomorphisms Rn → U(1). From that viewpoint, the Fourier transformation is aspecial case of harmonic analysis on locally compact abelian groups.

For j ∈ n, we compute

Djf(ξ) = ∫Rne−ixξDjf(x)dx = −∫

Rn−iiξje

−ixξf(x)dx

by partial integration, from which one sees that

Djf(ξ) = ξj f(ξ)and inductively

(3.1.8) Dαf(ξ) = ξαf(ξ)

For each Schwartz function f , one clearly has ∣f(ξ)∣ ≤ ∥f∥L1 . Because Djf

is again a Schwartz function, we find that ξj f(ξ) is a bounded function, and it-

eratively, we find that ξαf is a bounded function, for all α. Thus the Fouriertransform of a Schwartz function is rapidly decreasing. By the Lebesgue domi-

nated convergence theorem, the function f is differentiable (since the ξ-derivativeof the integrand is a Schwartz function and hence L1), and by differentiating underthe integral:


Dj f(ξ) = −xif(ξ).By induction, one finds that f is smooth and

(3.1.9) Dαf = (−1)∣α∣xαf(ξ).Therefore, the Fourier transform of a Schwartz function is a Schwartz function.

Example 3.1.10. The function h(x) = e−x2/2 is its own Fourier transform. One

checks this by differentiating the function eξ2/2h(ξ) using the rules just found and

the fact that h(0) = 1 (normalization of the integral!).

We introduce two scaling operators. Let f be a function on Rn. We set, for t > 0,

ft(x) ∶=1

tnf(x/t); f t(x) ∶= f(tx).

It is easy to check that

(3.1.11) ft = (f)t and f t = (f)t.

Proposition 3.1.12.

(1) If f, g ∈ S(Rn), then f ∗ g ∈ S(Rn).

(2) f ∗ g = f g.

Proof. The first part can be found in [46], p. 239. For the other part, compute

f ∗ g(ξ) = ∫ ∫ f(y)g(x − y)e−ixξdydx.

Since f and g are Schwartz functions, the integrand is in L1(R2n), and we canapply Fubini’s theorem and the change of variables y → y, x→ z + y to obtain

∫ ∫ f(y)g(z)e−i(z+y)ξdzdy = f(ξ)g(ξ).

For a Schwartz function f , we define f−(x) ∶= f(−x). It is straightforward toshow

f− = f−; (f ∗ g)− = f− ∗ g−; (fg)− = f−g−.

Theorem 3.1.13. (The Fourier inversion formula) For all Schwartz functions f ,

we haveˆf = f−. More explicitly, f(x) = ∫Rn e

ixξ f(ξ)dξ.

Proof. Let g ∈ S(Rn). We compute

∫ f(ξ)e−ixξg(ξ)dξ = ∫ ∫ f(y)e−i(y+x)ξg(ξ)dydξ = ∫ f(y)g(x + y)dy

since the integrand is in L1(R2n) (this was the purpose of introducing the inte-grating factor g). Inserting gt for g, we obtain, using 3.1.11

(3.1.14) ∫ f(ξ)e−ixξgt(ξ)dξ = ∫ f(y) 1

tng(x + y

t)dy y=tu−x= ∫ f(tu − x)g(u)du.

34 JOHANNES EBERT

Now specialize to the case g(ξ) = e−∣ξ∣2/2 and consider the limit t → 0. The

integrand on the right-hand side of 3.1.14 converges pointwise to f(−x)g, and isbounded by ∥f∥L∞g, and so by the dominated convergence theorem, the limit be-comes

limt→0

∫ f(tu − x)g(u)du = f(−x)∫ g(u)du = f(−x)

by the normalization of the Lebesgue measure. The integrand of the left-hand

side of 3.1.14 is bounded by the L1-function f , and gt(ξ) → 1 as t → 0. So by thedominated convergence theorem we obtain

limt→0

∫ f(ξ)e−ixξgt(ξ)dξ = ∫ f(ξ)e−ixξdξ = ˆf(x),

which was to be shown.

Corollary 3.1.15.

(1) fg = f ∗ g.

(2) The map f ↦ f is a bijective map S → S.

Proof. The second part is clear; since twice the Fourier transform is the reflectionmap f ↦ f− and so bijective. By the bijectivity, we find using 3.1.12

fg = f ∗ g⇔ fg = (f ∗ g)ˆ= ˆ

f ˆg = f−g−

which is true.

Recall that the L2-inner product is given by

⟨f, g⟩ ∶= ∫ ¯f(x)g(x)dx.

Theorem 3.1.16. (The Plancherel theorem) For all Schwartz functions f, g, wehave

⟨f, g⟩ = ⟨f , g⟩.

Proof. Let us momentarily denote (f, g) ∶= ⟨f , g⟩. The first step is

(3.1.17)

(f , g) = ∫ f(x)g(x)dx = ∫ ∫ f(ξ)e−ixξg(x)dξdx = ∫ f(ξ)g(ξ)dξ = (f, g)

where we used Fubini. Note that there are no conjugation signs. It is easy tosee that

f(ξ) = ˆf−(ξ).Compute, using 3.1.17,

(f , g) = (f , ˆg−) = ( ˆf, g−) = ((f)−, g−) = (f , g)and this proves the theorem.

Corollary 3.1.18. The Fourier transform extends to an isometry L2 → L2

This is clear because the Schwartz space lies dense in L2 (since it containsC∞c (Rn)). Warning: the defining formula for the Fourier transform only holds

for functions in L1 ∩L2.


3.2. Sobolev spaces in Rn.

Definition 3.2.1. On the Schwartz space, we introduce the Sobolev norm, for eachs ∈ R, by

∥f∥2W s ∶= ∫ (1 + ∣ξ∣2)s∣f(ξ)∣2dξ.

If it is clear that we mean the Sobolev norm (and not an Lp or Ck-norm), we write∥f∥s. The Sobolev space W s(Rn) is the completion of S(Rn) with respect to thes-norm. If U ⊂ Rn is open, we let W s(U) be the closure of C∞

c (U) ⊂ S(Rn). Wecan define Sobolev spaces of vector valued functions, using a scalar product on thevalue space.

We often also omit the domain of definition of the functions and write simplyW s.

Lemma 3.2.2.

(1) W 0 = L2.(2) For s ≥ t, we have ∥f∥t ≤ ∥f∥s and hence get a continuous map W s →W t.(3) This inclusion map is injective.

Proof. The first and second part are obvious (by the Plancherel theorem and be-cause the smooth functions with compact support lie dense in L2, so does theSchwartz space). For the third part, let fn be a ∥∥s-Cauchy sequence of Schwartz

functions and assume that ∥fn∥t → 0. Let gn ∶= ∣fn∣2(1+∣ξ∣2)t and let µ = (1+∣ξ∣2)s−t,which is a nonzero function.Taking the Fourier transforms and the definition of theSobolev norm, we obtain from our assumptions that

∥gn∥L1 → 0; ∥µgn − µgm∥L1 → 0.

Since µgn is an L1-Cauchy sequence, it converges almost everywhere (by [46],Theorem VI.5.2), and also gn converges almost everywhere, namely to 0. Since µgnis a Cauchy sequence, and its pointwise limit is 0, it follows that ∥µgn∥L1 → 0, by[46], Corollary VI.5.4.

Remark 3.2.3. We will use the third part only once, but in a crucial way: itis this statement that allows to go from arguments in the completion to actualfunctions. The use of the nontrivial relationship of L1-convergence with pointwiseconvergence is important. Viewing L1 in a purely abstract fashion as a completionis possible, but insufficient for the applications. If s > 0, we have the inclusionW s → L2, which allows us to consider elements of W s as actual functions. If s < 0,no such interpretation is possible. In fact, we will see soon that the famous Dirac δ-”function” is an element of W s for s << 0. The only way to reinterprete elements inW s for negative s is as distributions. However, the theory of distributions requiresmore background and we rather avoid it. This does not mean that the Sobolevspaces W s, s < 0, are irrelevant for us, but need to be treated with care.

Lemma 3.2.4. If s > 0 is an integer, the Sobolev norm is equivalent to the norm∥f∥s = ∑α≤s ∥Dαf∥L2 . Hence any differential operator P with constant coefficientsof order k induces a continuous map W s →W s−k.

Proof. There are constants c1, c2 > 0 such that for all x ∈ Rn

c1(1 + ∣x∣)s ≤ ∑∣α∣≤s

xα ≤ c2(1 + ∣x∣)s

36 JOHANNES EBERT

holds (this is the standard growth estimate for polynomials). Together withthe definition of the Sobolev norm, the rules for the Fourier transform and thePlancherel theorem, we get that the Soboloev norm is equivalent to the norm ∥f∥2 =∑α≤s ∥Dαf∥2

L2 . Then one uses the inequalities (true for all a1, . . . , ak ∈ R) ∑ki=1 a2i ≤

(∑ki=1 ∣ai∣)2 ≤ k∑ki=1 a2i which follow from the Cauchy-Schwarz inequality.

Example 3.2.5. Let L be the differential operator

Lf ∶= f −∆f = f +n

∑i=1

(Di)2f.

By the basic rules for the Fourier transform, we have

Lf(ξ) = (1 + ∣ξ∣2)f(ξ).Therefore, L ∶ S(Rn)→ S(Rn) is invertible with inverse M given by

Mf(ξ) ∶= (1 + ∣ξ∣2)−1f ;

moreover, the identity

(3.2.6) ∥Lf∥s = ∥f∥s+2

holds. This will be used later.

Theorem 3.2.7. (The Sobolev embedding theorem) Let s > k + n2

. Then there is a

constant C such that ∥u∥Ck ≤ C∥u∥s for all u ∈ S. Hence (since Ck is complete),we get a continuous inclusion W s → Ck.

Proof. Let ∣α∣ = l ≤ k and x ∈ Rn and u ∈ S. Then

∣Dαu∣(x) = ∣∫ ξαudξ∣ ≤ ∫ ∣ξ∣l∣u(ξ)∣(1 + ∣ξ∣2)s/2(1 + ∣ξ∣2)s/2dξ ≤

(∫ ∣u(ξ)∣2(1 + ∣ξ∣2)s)1/2(∫ ∣ξ∣2l(1 + ∣ξ∣2)−s)1/2 ≤ (∫ ∣ξ∣2l(1 + ∣ξ∣2)−s)1/2∥u∥s

by the Cauchy-Schwarz inequality, provided that ∣ξ∣l(1 + ∣ξ∣2)s/2 is an L2-function.But

∫ ∣ξ∣2l(1 + ∣ξ∣2)−s ≤ ∫∞

0∫Sn−1

r2l(1 + r2)−srn−1,

and this is finite if (and only if) 2l − 2s + n − 1 < −1, i.e. s > l + n2

.

Theorem 3.2.8. (The Rellich compactness theorem) Let U ⊂ Rn be relatively com-pact and s > t. Then W s(U)→W t is a compact operator.

Proof. We have to prove: if un is a sequence of smooth functions supported inU and if ∥un∥s ≤ 1, then a subsequence of un converges in the t-norm (think amoment on the definition of a compact operator to see why this is enough). Picka compactly supported function a such that a∣U ≡ 1, so that auk = uk for all k. Itfollows that uk = a ∗ uk and

Dj uk = (Dj a) ∗ uk.Thus we can estimate Dj uk(ξ) by


∣Dj a ∗ uk(ξ)∣ ≤ ∫ ∣Dj a(ξ − η)uk(η)dη∣ ≤

(∫ ∣Dj a(ξ − η)∣2(1 + ∣η∣2)−s))1/2(∫ ∣uk(η)∣2(1 + ∣η∣2)s)1/2 ≤ C∥uk∥sby Cauchy-Schwarz. The first integral exists since a is a Schwartz function. SoDj uk(ξ) is uniformly (in k!) bounded, and by a similar argument shows uk(ξ) isbounded. It follows that the family uk is equicontinuous, and Arzela-Ascoli providesa subsequence, also called uk, that converges uniformly on compact subsets of Rn.We claim that uk is a W t-Cauchy sequence. Let ε > 0. Let us compute

∥uk − ul∥2t = ∫∣ξ∣≤R

∣uk − ul∣2(1 + ∣ξ∣2)tdξ + ∫∣ξ∣≥R∣uk − ul∣2(1 + ∣ξ∣2)tdξ.

On the part ∣ξ∣ ≥ R, we estimate (1 + ∣ξ∣2)t ≤ (1 + ∣ξ∣2)s(1 +R2)t−s. Thus

∫∣ξ∣≥R∣uk − ul∣2(1 + ∣ξ∣2)tdξ ≤ (1 +R2)t−s(∥uk∥2

s + ∥ul∥2s) ≤ 2(1 +R2)t−s

and since t−s < 0, we can make this term smaller than ε/2, by choosing R sufficientlylarge. Because uk is uniformly convergent on ∣ξ∣ ≤ R and (1 + ∣ξ∣2)t is boundedon this set, the integral

∫∣ξ∣≤R∣uk − ul∣2(1 + ∣ξ∣2)tdξ

converges to zero as k, l →∞.

We already said several times that Hilbert spaces are self-dual via the scalarproduct, and the Sobolev spaces are Hilbert spaces. However (at the moment thisis not yet clear), the actual norm on the Sobolev space is negotiable (it is onlya ”Hilbertian space” in Bourbakian rigor). Only when we consider operators onRiemannian manifolds, the scalar product on L2 will have an intrinsic meaning.Nevertheless, the self-duality is important for Sobolev spaces, and it takes the formof a perfect pairing W s ×W −s → C. Often, a statement is easier to prove on oneside of the Sobolev chain, and duality allows us to pass to the other side.

Proposition 3.2.9. (Duality) The sesquilinear form1 S×S → C, (f, g)↦ ∫ f(x)g(x)dxextends to a pairing W s ×W −s → C and satisfies

(1) ∣(f, g)∣ ≤ ∥f∥s∥g∥−s.(2) ∥f∥s = supg≠0

∣(f,g)∣∥g∥−s = sup∥g∥−s≤1 ∣(f, g)∣.

(3) The induced map W −s → (W s)′, g ↦ (f ↦ (f, g)) is an isometric isomor-phism.

Proof. By Plancherel and Cauchy-Schwarz:

(f, g) = (f , g) = ∫Rnf(ξ)(1 + ∣ξ∣2)s/2g(ξ)(1 + ∣ξ∣2)−s/2dξ ≤ ∥f∥s∥g∥−s.

The inequality sup∥g∥−s≤1 ∣(f, g)∣ ≤ ∥f∥s follows immediately. If f is a Schwartz

function, define g by g = f(1 + ∣ξ∣2)s. Then (f, g) = (f , g) = ∥f∥2s by the definition

of the Sobolev norm and the Plancherel theorem. Moreover ∥g∥2−s = ∥f∥2

s, in otherwords

1Before, we have denoted this by ⟨f, g⟩. The reason for the notation switch is that here thepairing plays a different role.

38 JOHANNES EBERT

∣(f, g)∣∥g∥−s

= ∥f∥s

and this proves the other inequality.For the third part, it follows easily from what we already proved that φ ∶W −s →

(W s)∗ is norm-preserving. Thus it has closed image. What we have to show is thefollowing claim

If V is a Hilbert space, and H ⊂ V ∗ a closed subspace, such that for all

v ∈ V , we have ∥v∥ = supx∈H∣x(v)∣∥x∥ . Then H = V ∗.

To prove the claim, translate it using the self-duality of Hilbert spaces to thefollowing statement:

H Hilbert space, W ⊂ H closed, such that ∥x∥H = supw∈W,∥w∥=1 ∣(x,w)∣.Then W =H.

This is easy. Assume that W ≠ 0 and pick an element in W of norm 1 to geta contradiction.

3.3. The fundamental elliptic estimate. In this subsection, we will prove thefollowing two results.

Proposition 3.3.1. Let P a differential operator of order k, with compact support.Then for each s ∈ Z, P induces a bounded operator P ∶ W s → W s−k. Moreover,if the coefficients of P depend smoothly on a parameter t ∈ R, then t ↦ Pt is acontinuous map R→ Lin(W s,W s−k).

The other result is of fundamental importance for the proof of the regularitytheorem.

Theorem 3.3.2. (Garding inequality) Let P be a differential operator of order kon Rn. Let U ⊂ Rn be relatively compact and assume that P is elliptic over U .Then there exists a constant C, depending on P , U and s ∈ Z, such that for eachu ∈ C∞

c (U) with support in U , the elliptic estimate

∥u∥s ≤ C(∥u∥s−k + ∥Pu∥s−k)holds.

The proofs of both results rely on an estimate for the multiplication operatorf ↦ af on S, when a has compact support. We will show below that for eacha ∈ C∞

c (Rn), multiplication by a induces a continuous map W s → W s, for eachs ∈ Z. Together with Lemma 3.2.4, this proves the result. However, we need twomore precise estimates on the operator norm of D. The first estimate is used toshow that if the coefficients of D depend smoothly on a parameter, then the inducedoperators depend continuously on the parameter. This will eventually prove thatthe indices of two operators whose symbols are homotopic will agree. The secondestimate will be used in the proof of Gardings inequality.

Proposition 3.3.3. Let a ∈ C∞c . Then f ↦ af extends to a bounded operator

Ma ∶W s →W s, for each s ∈ Z. More precisely, we have the following estimates:

(1) ∥au∥s ≤ C∥a∥C ∣s∣∥u∥s, for all s ∈ Z and a constant C independent of a.(2) ∥au∥s ≤ ∥a∥C0∥u∥s +C(a)∥u∥s−1. In other words, the ”leading term” can be

estimated by the C0-norm of a, with a ”lower perturbation”, whose normdepends on higher derivatives of a.


Proof. For nonnegative k, we compute

∥au∥2k = ∑

∣α∣≤k∥Dα(au)∥2

0 ≤ ∑∣α∣≤k

∑β+γ=α

α!

γ!β!∥(Dβa)(Dγu)∥2

0.

We can estimate

∥(Dβa)(Dγu)∥0 ≤ ∥Dβa∥C0∥Dγu∥0 ≤ ∥a∥C ∣β∣∥u∥W ∣γ∣ .

Thus we find

∥au∥k ≤ ∥a∥C0∥u∥k +C∥a∥Ck∥u∥k−1.

This is the second estimate for k ≥ 0, the constants C depends on k and n alone.To get the first estimate, we estimate further

∥a∥C0∥u∥k +C∥a∥Ck∥u∥k−1 ≤ C ′∥a∥Ck∥u∥k,obtaining the first estimate for positive k. For −k, the estimate follows by duality:

∥af∥−k = sup∥g∥k≤1

∣⟨af, g⟩∣ = sup∥g∥k≤1

∣⟨f, ag⟩∣ ≤ sup∥g∥k≤1

∥f∥−k∥ag∥k ≤ sup∥g∥k≤1

∥f∥−k∥a∥Ck∥g∥k.

Before we prove the second estimate for negative values of k, we note a corollaryof the first estimate.

Corollary 3.3.4. Let P be a differential operator with compact support, of orderk. Then, for each s ∈ Z, there is a constant C = C(P, s) such that ∥Pu∥s−k ≤ C∥u∥sholds. The constant C can be bounded by the sum of the Cl-norms of the coefficientsof P , with l = ∣s − k∣.

If I = (t0, t1) ⊂ R, and if Pt is a family of order k differential operators thatdepend smoothly on t, then I → Lin(W s,W s−k), t↦ Pt is continuous.

Proof. By Proposition 3.3.3, one estimates

∥Pu∥s−k ≤ ∑∣α∣≤k

∥AαDαu∥s−k ≤ ∑∣α∣≤k

∥Aα∥C ∣s−k∣∥u∥s−∣α∣.

This proves the first assertion. The second is an easy consequence, as the dif-ferentiability assumption on Pt shows that the derivatives of the coefficients of Ptdepend continuously on t.

End of the proof of Proposition 3.3.3. We have to prove the estimate ∥au∥s ≤ ∥a∥C0∥u∥s+C(a)∥u∥s−1 for negative integers s and do so by downwards induction on s; the cases ≥ 0 has been settled before. We make use of the operator L discussed in 3.2.5 andthe fact that L is an isometry W s+2 → W s, for all s. Namely, we estimate (underthe assumption that the proof has been given for all t > s)

∥aLu∥s ≤ ∥[a,L]u∥s + ∥L(au)∥s ≤ C(a)∥u∥s+1 + ∥L(au)∥s =(by Corollary 3.3.4)

= C(a)∥Lu∥s−1 + ∥au∥s+2 ≤(by 3.2.6)

40 JOHANNES EBERT

≤ C(a)∥Lu∥s−1 + ∥a∥C0∥u∥s+2 +C ′(a)∥u∥s+1 ≤ C ′′(a)∥Lu∥s−1 + ∥a∥C0∥Lu∥s.Since L is an isomorphism, this finishes the proof.

Corollary 3.3.5. Let P be a differential operator with compact support of order kand s ∈ Z. Then, for all u ∈W s,

∥Pu∥s−k ≤ C1∥u∥s +C2∥u∥s−1,

where the constant C1 can be bounded by the sum of the C0-norms of the coeffi-cients of P , and C2 does depend on the higher derivatives of the coefficients of P(but not on u).

Proof. We can write P = ∑∣α∣=kDαaα + Q with an operator of order k − 1. By

Corollary 3.3.4, ∥Qu∥s−k ≤ C∥u∥s−1 for some constant C = C(Q) depending on thecoefficients of Q. On the other hand, by the second estimate of Proposition 3.3.3,

∥Dαaαu∥s−k ≤ ∥aαu∥s ≤ ∥u∥s∥aα∥C0 +C∥u∥s−1

with C = C(a) depending on a and its derivatives. Putting both estimatestogether, we obtain the claimed estimate.

For the proof of the Garding inequality, we need another preliminary fact.

Lemma 3.3.6. (Peter and Paul estimate) Let r < s < t ∈ R. Then for each ε > 0,there exists a C(ε) > 0 such that for all u ∈ S, the estimate

∥u∥s ≤ ε∥u∥t +C(ε)∥u∥rholds.

Proof. For y ≥ 1, the inequality

1 ≤ yt−s + (1/y)s−r

holds because either y or 1/y is ≥ 1 and both exponents are positive. For y =(1 + ∣ξ∣2)ε 1

t−s , we obtain

1 ≤ (1 + ∣ξ∣2)t−sε + (1 + ∣ξ∣2)r−sεr−st−s

or

(1 + ∣ξ∣2)s ≤ (1 + ∣ξ∣2)tε + (1 + ∣ξ∣2)rεr−st−s

which implies the claim by integration.

Proof of Theorem 3.3.2. The proof is a prototypical example of a ”local-to-global”argument in analysis. We proceed in three steps:

(1) P has constant coeffients. This is much easier (that this is so is one of thetwo reasons why the whole proof of the local regularity theorem is muchsimpler for the classical operators on Rn, as the Cauchy-Riemann operatorand the Laplace operator).

(2) P has variable coefficients, but the functions are required to have smallsupport.

(3) The general case.


First step: the coefficients are constantRecall that P can be represented by a function p(x, ξ) which is smooth in the

x-variable and a degree k polynomial in the ξ-variable (and has values in matrices).In the first step, we assume that p(x, ξ) = p(ξ) does not depend on x, in other

words, P has constant coefficients. Ellipticity states that there exist c, R > 0 suchthat ∣p(ξ)∣ ≥ c(1 + ∣ξ∣2)k/2 for all ∣ξ∣ ≥ R. Recall that in the Fourier picture, theoperator is written as

P u(ξ) = p(ξ)u(ξ).Now

∥u∥2s = ∫Rn−BR(0)

∣u(ξ)∣2(1 + ∣ξ∣2)s + ∫BR(0)

∣u(ξ)∣2(1 + ∣ξ∣2)sdξ.

The second summand estimates as

∫BR(0)

∣u(ξ)∣2(1 + ∣ξ∣2)sdξ ≤ sup∣ξ∣≤R

(1 + ∣ξ∣2)k ∫BR(0)

∣u(ξ)∣2(1 + ∣ξ∣2)s−kdξ ≤ C0∥u∥2s−k

with C0 = (1 +R2)k. The first summand is

≤ ∫Rn−BR(0)

∣p(ξ)u(ξ)∣2 1

c(1 + ∣ξ∣2)s−kdξ ≤ 1

c∥Pu∥2

s−k

and this settles the case of constant coefficients (the restriction on the supportsof u was not necessary, and it works for each s ∈ R).

The general case, local versionLet x0 ∈ U . We claim that there exists a neighborhood V ⊂ U of x0 and a constant

C = C(x0, s) such that for all u ∈ C∞c (V ), the estimate ∥u∥s ≤ C(∥u∥s−k + ∥Pu∥s−k)

holds.Let δ > 0. Let P0 be the differential operator with constant coefficients associated

with p0(ξ) ∶= p(x0, ξ). As we assumed that P is elliptic over U , the operator P0 iselliptic. Now pick a neighborhood W of x0 such that all coefficients of P − P0 arebounded by δ on W . Moreover, we pick x0 ∈ V ⊂ V ⊂ W and a bump function λthat is 1 on V and has support in W . If supp(u) ⊂ V , then by the first part of theproof (and the triangle inequality), we obtain

∥u∥s ≤ C1(∥u∥s−k + ∥P0u∥s−k) ≤ C1(∥u∥s−k + ∥(P0 − P )u∥s−k + ∥Pu∥s−k)

with C1 depending on x0 alone. We now care about the summand ∥(P0−P )u∥s−k.Observe that

∥(P0 − P )u∥s−k = ∥(P0 − P )λu∥s−ksince λu = u; the operator (P0 − P )λ has compact support and the C0 norm of

all coefficients is bounded by δ. By Corollary 3.3.5, we find

∥(P0 − P )λu∥s−k ≤ δC2∥u∥s +C3∥u∥s−1,

with a universal constant C2 and C3 depending on P and δ, because the higherderivatives of the bump function λ become large when δ is small. Let ε > 0 andassume k > 1 for the moment. By the Peter-Paul estimate, we find C4 such that

42 JOHANNES EBERT

C1C3∥u∥s−1 ≤ ε∥u∥s +C4∥u∥s−k.Putting everything together, we obtain

∥u∥s ≤ (C1 +C4)∥u∥s−k + (δC1C2 + ε)∥u∥s +C1∥Pu∥s−k),and note that all constants except C4 are independent of δ and ε (C4 depends

on both; C2 is universal and C1 only depends on x0). If k = 1, the above estimateis still true even with ε = 0, without appealing to the Peter-Paul inequality. Pick δand ε small enough so that δC1C2 + ε < 1. Thus

(1 − δC1C2 + ε)∥u∥s ≤ (C1 +C4)∥u∥s−k +C1∥Pu∥s−k).Dividing by (1 − δC1C2 + ε) finishes the second step.General case, global versionThe third step deals with the general case. There exists a larger open W ⊃ U so

that P is elliptic over W . Cover U by finitely many V1, . . . Vm ⊂W as found in thesecond part, such that there is a constant Ci with

∥u∥s ≤ Ci(∥u∥s−k + ∥Pu∥s−k)whenever supp(u) ⊂ Vi. Let C ∶= maxiCi. Pick a finite partition of unity µ1, . . . , µmsubordinate to the cover by the Vi’s. For general u with support in U , we conclude

∥u∥s ≤m

∑i=1

∥µiu∥s ≤m

∑i=1

C(∥µiu∥s−k + ∥Pµiu∥s−k) ≤ C ′∥u∥s−k +Cm

∑i=1

∥Pµiu∥s−k),

the constant C ′ depending on C and the C ∣s−k∣-norm of the functions µi. We canestimate

∥Pµiu∥s−k ≤ ∥[P,µi]u∥s−k + ∥µiPu∥s−k ≤ C ′′i ∥u∥s−1 +C ′′′

i ∥Pu∥s−kbecause [P,µi] has order k − 1 and thus

∥u∥s ≤ C ′∥u∥s−k +C ′′∥u∥s−1 +C ′′′∥Pu∥s−k.By Peter and Paul, C ′′∥u∥s−1 ≤ ε∥u∥s +C(ε)∥u∥s−k and picking ε < 1 finishes the

proof.

3.4. A smoothing procedure. We will use Gardings inquality for the proof ofthe regularity theorem. It states that if f ∈ W s and u ∈ W s and if Pu = f , thenin fact u ∈ W s+k. By passage to the completion, Gardings inequality continues tohold; thus we have ∥u∥s+k ≤ C(∥u∥s + ∥Pu∥s) for all u ∈ W s. Gardings inequalitygives an a priori estimate for a solution u of Pu = f : ∥u∥s+k ≤ C(∥u∥s + ∥f∥s),suggesting that u is always in W s+k if f ∈W s. If f is smooth, this should give thatu is smooth. However, the assumption requires that u is already in W s+k, and weare going around in a circle. To get around this problem, we introduce now theFriedrichs mollifiers.

Let φ ∈ C∞c (Rn) be a function with φ ≥ 0, ∫ φ(x)dx = 1 and φ(−x) = φ(x). For

ε > 0, we let φε(x) = 1εnφ(x

ε).

Definition 3.4.1. The Friedrichs mollifier is the operator Fε ∶ S → S; u↦ φε ∗ u.


Proposition 3.4.2.

(1) Fε extends to a bounded operator W s →W s, with operator norm ≤ 1.(2) Fε commutes with all differential operators with constant coefficients.(3) For each u ∈W s, Fεu is in C∞ ∩W s.(4) For each u ∈W s, Fεu→ u in the W s-norm.

Proof. For each a ∈ C∞c (Rn) and u ∈ S(Rn), one estimates

∥a ∗ u∥2s = ∫ ∣a ∗ u(ξ)∣2(1 − ∣ξ∣2)sdξ = ∫ ∣a(ξ)u(ξ)∣2(1 − ∣ξ∣2)sdξ ≤ ∥a∥2

L∞∥u∥2s.

But ∥a∥L∞ ≤ ∥a∥L1 and since ∥φε∥L1 = ∥φ∥L1 , the proof of (1) is complete. Part(2) is an easy consequence of Proposition 3.1.3.

For part (3), consider first the case s ≥ 0. Since W s ⊂ L2, Fεu is smooth byProposition 3.1.3; one has to use that smoothness is a local property, and if u ∈ L2

and x ∈ Rn, then Fεu(x) = Fε(µu)(x) for some cut off function µ with large support;but µu is L1. For negative s, suppose that part (3) has been proven for the values. Any u ∈W s−2 can be written uniquely as Lv, v ∈W s. Then

Fε(Lv) = LFεvand this is smooth.For part (4), let u ∈W s and pick v ∈ C∞

c with ∥u − v∥s < δ/3, so that

∥u − Fεu∥s ≤ ∥u − v∥s + ∥v − Fεv∥s + ∥Fε(v − u)∥s ≤ ∥v − Fεv∥s +2

3δ.

But

∥v − Fεv∥2s = ∫ ∣(φε − 1)v∣2(1 + ∣ξ∣2)sdξ = ∫ ∣(φ(εξ) − 1)∣2∣v(ξ)∣2(1 + ∣ξ∣2)sdξ

and the integrand is pointwise convergent to 0, and bounded by the L1-function2∣v∣(1 + ∣ξ∣2)s, and so the integral tends to zero by the dominated convergencetheorem.

Proposition 3.4.3. Let U ⊂ Rn be relatively compact, let u ∈ W r(U), s > r andassume that ∥Ftu∥s ≤ C uniformly in t. Then u ∈W s.

Proof. Let Fn ∶= Ftn where tn → 0. Let Λn ∶ W −s → C be the functional v ↦⟨Fnu, v⟩. We have ∣Λn(v)∣ ≤ C∥v∥−s, by Proposition 3.2.9. So the family Λn isequicontinuous and bounded, by 1.4.5. Thus, by Arzela-Ascoli, there is a subse-quence, also denoted Λn, such that Λn converges uniformly on all compact subsetsof W −s, to some linear functional Λ ∶ W −s → C which is also bounded by C. ByProposition 3.2.9, there is w ∈ W s such that Λ(v) = (w, v), for all v ∈ W −s. Weclaim that the image of w in W r is equal to u.

By Rellich, the image of B1(W −r(U ′)) in W −s is relatively compact for all rel-atively compact U ′, and so Λn → Λ in (W −r(U ′))∗. We assumed that u ∈ W r(U)and hence Fnu → u in W r. So the restriction of Λ to W −r(U ′) must be given bypairing with u, for each relatively compact U ⊂ U ′ ⊂ Rn. The union of the Sobolevspaces W −r(U ′) over all U ′ is dense in W −r, and so the restriction of Λ to W −r isgiven by pairing with u. Hence u = w and the proof is complete.

44 JOHANNES EBERT

Remark 3.4.4. This proposition can be formulated more abstractly, using thenotion of weak convergence.

Proposition 3.4.5. (Friedrich’s lemma) Let P a differential operator of orderk ≥ 1 with compact support and let Ft be a family of Friedrich mollifiers. Then thecommutator [Ft, P ] is a bounded operator W s → W s−k+1 for each s ∈ R, and theoperator norm is uniformly bounded (i.e. independent of t).

For the proof, we need a useful estimate:

Lemma 3.4.6. (Peetre inequality) Let x, y ∈ Rn and s ∈ R. Then

(1 + ∣x∣2)s

(1 + ∣y∣2)s≤ 2∣s∣(1 + ∣x − y∣2)∣s∣.

Proof. By switching the roles of x and y, it is enough to consider s ≥ 0, and moreovers = 1. But

(1 + ∣x∣2) = 1 + ∣x − y∣2 + ∣y∣2 + 2(x − y)x ≤ 1 + ∣x − y∣2 + ∣y∣2 + (∣x − y∣2 + ∣y∣2) ≤

2(1 + ∣y∣2 + ∣x − y∣2 + ∣y∣2∣x − y∣2) = 2(1 + ∣y∣2)(1 + ∣x − y∣2).

Proof of Friedrichs lemma. (This proof is the solution of an exercise in [65], p.235). The result is proven by induction on the order k of P . The case k = 1contains the core argument. For higher order, one argues by induction on k, inan algebraic way. Namely, let P be a differential operator of order k and ∂ be aconstant coefficient operator of order 1. Then, using that ∂ commutes with Ft, onegets [P∂,Ft] = [P,Ft]∂, which easily implies the inductive step, for operators ofthe form P∂. But by the very definition of a differential operator, each operator oforder k + 1 is the sum of such special operators.

Now consider the case k = 1. If the principal symbol of P is zero, the proof istrivial, since Ft itself is uniformly bounded and P has order zero. So we are let tostudy the operator cDj , for some smooth, compactly supported function c (eachorder one operator can be written as a sum of such operators, plus an order 0 term).First, we need the estimate for ξ, η ∈ Rn:

(3.4.7) ∣(ξj + ηj)φ(t(ξ + η)) − ξj φ(tξ)∣ ≤ C ∣η∣for a constant C that does not depend on t. To see this, we write the term to

be estimated as the absolute value of

(ξj + ηj)φ(t(ξ + η)) − ξj φ(tξ) = ∫1

0

∂

∂s((ξj + sηj)φ(t(ξ + sη)))ds =

= ∫Rnφ(x)∫

1

0

∂

∂s((ξj + sηj)e−ixt(ξ+sη))dsdx =

∫Rnφ(x)∫

1

0ηje

−ixt(ξ+sη)dsdx + ∫Rnφ(x)∫

1

0(ξj + sηj)(−itxη)e−ixt(ξ+sη)dsdx.

The first integral is bounded by ∫ φ(x)dx∣η∣ = ∣η∣. The second one equals


∫1

0∫Rnφ(x)(xη) ∂

∂xje−ixt(ξ+sη)dsdx = −∫

1

0∫Rn

∂

∂xj(φ(x)(xη))e−ixt(ξ+sη)dsdx

by partial integration. The absolute value can be estimated by

∫1

0∫Rn

∣η∣∣ ∂∂xj

(φ(x)x)∣dsdx ≤ ∣η∣∫Rn

∣ ∂∂xj

(φ(x)x)∣dx.

So the proof of 3.4.7 is complete.Now consider the differential operator Pu = cDju. The commutator is

[Ft, P ] = FtcDj − cDjFt = FtcDj − FtDjc + FtDjc − cDjFt = Ft[c,Dj] + [FtDj , c]

using that Ft commutes with constant coefficient operators. As [c,Dj] is of order0, Ft[c,Dj] is uniformly bounded, and we only have to take care of [FtDj , c]. Foru ∈W s and v ∈W −s, we have by Plancherel’s theorem

⟨[FtDj , c]u, v⟩ = ⟨([FtDj , c]u)ˆ, v⟩.This is equal to

∫Rn

(φt(ξ)ξj(c ∗ u)(ξ) − (c ∗ φtMj u)(ξ)v(ξ)dξ,

by the rules for the Fourier transform, where Mj stands for multiplication bythe function ξj . Writing the convolution out and using 3.1.11, we obtain

∫ ∫ (φ(tξ)ξj c(η)u(ξ − η) − c(η)φ(t(ξ − η))(ξj − ηi)u(ξ − η)) ¯v(ξ)dξdη =

= ∫ ∫ c(η)¯v(ξ)u(ξ − η) (φ(tξ)ξj − φ(t(ξ − η))(ξj − ηi))dξdη.

By 3.4.7, the absolute value of this integral can be estimated by

C ∫ ∫ ∣c(η)¯v(ξ)u(ξ − η)∣∣η∣dξdη η→ξ−ζ= C ∫ ∫ ∣c(ξ − ζ)¯v(ξ)u(ζ)∣∣ξ − ζ ∣dξdζ =

= C ∫ ∫ ((1 + ∣ξ − ζ ∣2)s/2∣c(ξ − ζ)∣∣ξ − ζ ∣) (1 + ∣ξ − ζ ∣2)−s/2∣¯v(ξ)u(ζ)∣dξdζ =∶

C ∫ ∫ q(ξ − ζ)(1 + ∣ξ − ζ ∣2)−s/2∣¯v(ξ)u(ζ)∣dξdζ

By the Peetre inequality, this is estimated by (since s ≥ 0):

≤ C ∫ ∫ q(ξ − ζ)(1 + ∣ζ ∣2)s/2∣u(ζ)∣(1 + ∣ξ∣2)−s/2∣¯v(ξ)∣dξdζ.

(We use the symbol C for a constant that changes from line to line; the actualvalue of C is irrelevant for us) Using Cauchy-Schwarz, this is

≤ C (∫ ∫ q(ξ − ζ)(1 + ∣ζ ∣2)s∣u(ζ)∣2dζdξ)1/2

(∫ ∫ q(ξ − ζ)(1 + ∣ξ∣2)−s∣v(ξ)∣dζdξ)1/2

.

The first factor is

46 JOHANNES EBERT

≤ (∥u∥2s ∫ q(ξ)dξ)

1/2≤ C∥u∥s

and likewise the second factor is bounded by C∥v∥−s. Altogether, we get that

∣⟨[FtDj , c]u, v⟩∣ ≤ C∥u∥s∥v∥−sand by duality, we conclude that ∥[FtDj , c]u∥s ≤ C∥u∥s, the constant C not

depending on t and u. This concludes the proof.

3.5. Local elliptic regularity.

Theorem 3.5.1. (The local regularity theorem) Let P be a differential operatorof order k that is elliptic over U , U ⊂ Rn relatively compact. Let k, l be integers,f ∈ W l, and u ∈ W r. Assume that Pu = f (this equation takes place in W r−k).Then for each function µ ∈ C∞

c (U), µu ∈W l+k.

Corollary 3.5.2. Let u ∈ W r and P elliptic over supp(u). Assume that Pu issmooth. Then u is smooth over U .

This follows from the theorem by the Sobolev embedding theorem.

Proof of Theorem 3.5.1. By induction, we can assume that µu ∈ W k+l−1 and wehave to prove that µu ∈W k+l. By Garding’s inequality

∥Fεµu∥k+l ≤ C(∥Fεµu∥l+∥PFεµu∥l) ≤ C(∥Fεµu∥l+∥[P,Fε]µu∥l+∥Fε[P,µ]u∥l+∥FεµPu∥l).

Now all four summands are uniformly bounded (independent of ε):

∥Fεµu∥l ≤ ∥µu∥l ≤ C(µ)∥u∥l (by 3.4.2 and 3.3.3). ∥[P,Fε]µu∥l ≤ C∥µu∥l+k−1 ≤ C ′∥u∥l+k−1 by Friedrich’s lemma and 3.3.3. ∥Fε[P,µ]u∥l ≤ ∥[P,µ]u∥l by 3.4.2. Moreover, [P,µ] is an operator of orderk − 1 with compact support and so, by 3.3.4, ∥[P,µ]u∥l ≤ C∥u∥l+k−1.

∥FεµPu∥l ≤ ∥µf∥l by 3.4.2.

Appealing to Proposition 3.4.3 concludes the proof.

3.6. Sobolev spaces on manifolds. We now move on to globalize the theory sofar developed. We have to define Sobolev spaces on manifolds, and we will do thisonly for integral indices, as this is everything we need. From now on, M will alwaysbe a closed n-dimensional manifold and E →M a complex vector bundle. We picka finite cover of M by sets Ui with charts hi ∶ Ui ≅ Rn. Moreover, we pick bundletrivializations φi of E∣Ui and a partition of unity µi subordinate to the cover Ui.We define the Sobolev norm of u ∈ Γ(M,E) by

∥u∥2k ∶=∑

i

∥µiφi u (hi)−1∥2k.

As expected, the Sobolev space W s(M ;E) is defined to be the completion ofΓ(M ;E) with respect to that norm. We will then transfer the most importantresults from the previous sections to the manifold case. To get things started, oneneeds some pieces of information. The key is the following lemma.


Lemma 3.6.1. Let φ ∶ U ′ → V ′ be a diffeomorphism of open subsets of Rn andlet U ⊂ U ′ and V = φ(U) ⊂ V ′ be relatively compact. Then u ↦ u φ extends to abounded map W s(V )→W s(U), for all s ∈ Z.

Proof. Assume first s = k ∈ N and u ∈ C∞c (V ). Compute

∥u φ∥2k = ∑

∣α∣≤k∫ ∣Dα(u φ)(x)∣2dx.

Now a qualitative version of the chain rule in several variables for higher orderderivatives states that

Dα(u φ) = ∑∣β∣≤∣α∣

((Dβu) φ)Pα,β(φ)

where P is a universal polynomial in the derivatives of φ up to order ∣α∣ (anexplicit formula is the so-called Faa di Bruno formula). Therefore, because U andV are relatively compact,

∫ ∣Dα(uφ)(x)∣2dx ≤ C ∑∣β∣≤∣α∣

∫ ∣(Dβu)φ∣2dx = ∫ C ∑∣β∣≤∣α∣

∫ ∣(Dβu)∣2∣detDφ∣−2dy ≤ C ′∥u∥2k.

This settles the case of nonnegative k. For −k, we use duality. Let U ⊂ U ′′ ⊂ U ′

be an intermediate relatively compact subset and V ′′ = φ(U ′′). Observe that

∫U ′′

u(φ(x)) ¯g(x)dx = ∫V ′′

u(y)g(φ−1(y))∣detDφ(y)∣−1dy

and therefore, by duality,

∥u φ∥−k = supg∈C∞

c (U ′′),∥g∥k≤1∫U ′′

u(φ(x))g(x)dx =

supg∈C∞

c (U ′′),∥g∥k≤1∫V ′′

u(y)g(φ−1(y))∣detDφ(y)∣−1dy ≤ ∥u∥−k∥(g φ−1)∣detDφ∣−1∥k ≤

≤ C∥u∥−k∥g φ−1∥k ≤ C ′∥u∥−k∥g∥k.This finishes the proof.

Lemma 3.6.2.

(1) The equivalence class of the norm ∥∥k does not depend on the choices made.(2) If M has a distinguished Riemann metric and the bundle E a distinguished

hermitian bundle metric, then ∥∥0 is equivalent to the L2-norm defined in2.3.3.

(3) If u has support in a coordinate neighborhood Ui, then ∥φi u h−1i ∥k,Rn ≤

C∥u∥k,M , where C depends on the choice of the trivializations and charts.

Proof. The first claim follows from Lemma 3.6.1 and Proposition 3.3.3. The secondpart is similar, one uses local trivializations that respect the inner product. Thethird part is an easy exercise.

Using this lemma, we can transfer the known results to manifolds. Here is thesummary:

Theorem 3.6.3. Let M be a closed manifold and E → M be a hermitian vectorbundle. Then:

48 JOHANNES EBERT

(1) The inclusion map W l →W k is injective for k > l.(2) (Sobolev embedding) The elements of W l are Ck-sections, provided that

l > n2+ k.

(3) (Rellich compactness) The inclusion W l →W k is compact if l > k.(4) Each differential operator P of order k induces a continuous operator W l+k →

W l for all l. If P depends smoothly on a parameter t ∈ R, then t ↦ Pt is acontinuous map R→ Lin(W k+l,W l).

(5) (Gardings inequality) If P is elliptic, then there is a constant C such that∥u∥k+l ≤ C(∥u∥l + ∥Pu∥l) holds for all u ∈W k+l.

(6) (Duality) The map W k(M,E) → (W −k(M ;E))∗ given by u ↦ (v ↦ ⟨u, v⟩)is an antilinear isomorphism of Hilbert spaces.

No new ideas are required for the proof. As a sample, we show how to prove theGarding inequality. We use the notation from the beginning of this section. Denoteui ∶= φi u (hi)−1, so that

∥u∥k+l ≤ C∑i

∥µiui∥k+l ≤∑i

CCi(∥µiui∥l + ∥Pµiui∥l)

from Garding’s inequality in Rn. Using the third part of 3.6.2, we get

CCi(∥µiui∥l ≤ C ′∥u∥l.Let ai be a compactly supported function in Ui with aiµi = µi. Note that

µiPai = aiµiP = µiP . We obtain

∥Pµiui∥l ≤ ∥[P,µi]aiui∥l + ∥µiPaiui∥l ≤ Ci∥aiui∥l+k−1 +Ci∥µiPui∥lThe second is at most ∥Pu∥l by the definition of the Sobolev norm, and the first

summand can be estimated by the Peter-Paul inequality (in Rn) against ε∥aiu∥l+k+C∥aiu∥l. By Lemma 3.6.2, both summands are bounded by the respective Sobolevnorm.

Since the covering was finite (!!), the proof is completed by putting everythingtogether.

3.7. Global regularity and the Hodge theorem. Now we finally come to theproof of the Hodge decomposition theorem and the Fredholm property of ellipticoperators on closed manifolds. Let M be a closed manifold and let P ∶ Γ(M,E0)→Γ(M,E1) be an elliptic differential operator. We first globalize the regularity the-orem.

Theorem 3.7.1. Let Pu = f , f ∈W l, u ∈W r, for some integer r. Then u ∈W l+k.

Proof. Let U ⊂ M a coordinate chart. Let V ⊂ U be relatively compact. Pickfunctions µ,λ ∈ C∞

c (U) with µ ≡ 1 on V and µλ = µ. Since

µf = µPu = µP (λu)and µP is elliptic over V , the local regularity theorem 3.5.1 tells us that for each

ϕ ∈ C∞c (V ), the function ϕλu = ϕu belongs to W l+k. Cover M by finitely many

such sets Vi and let (ϕi)i be a partition of unity subordinate to this covering. Thusu = ∑i ϕiu belongs to W k+l.

For the proof of the Fredholm property, we need an abstract functional-analyticresult.


Proposition 3.7.2. Let U,V,W be Hilbert spaces, P ∶ U → V bounded and K ∶U →W compact. Assume that there exists a constant C with

∥u∥U ≤ C(∥Pu∥V + ∥Ku∥W )Then the kernel of P is finite dimensional and P has closed image.

Proof. Let un be a sequence with Pun = 0 and ∥un∥U ≤ 1. Then

∥un − um∥U ≤ C∥K(un − um)∥Wand by the compactness of K, Kun is subconvergent2. Therefore, a subsequence

un is a Cauchy sequence. This shows that each bounded sequence in the kernel ofP is subconvergent; and therefore ker(P ) is finite-dimensional.

To prove that the image of P is closed, it is enough to consider P ∣ker(P ) , inother words, we can assume that P is injective.

We claim that there exists a c > 0 with c∥u∥ ≤ ∥Pu∥ for all u ∈ U . Suppose thisis not the case; i.e. for each b > 0, there is u with ∥u∥ = 1 and ∥Pu∥ ≤ b. We canthen find a sequence un such that ∥un∥ = 1 and ∥Pun∥→ 0.

Since K is compact, we can assume without loss of generality that Kun is con-vergent. Therefore

∥un − um∥U ≤ C(∥P (un − um)∥V + ∥K(un − um)∥W )which converges to 0. Thus un is a Cauchy sequence and the limit u must have

∥u∥U = 1 (since all un have norm 1) and Pu = 0 (since Pun → 0 and P is continuous).This contradicts the assumption that P is injective, and this contradiction provesthe existence of the constant c.

Now let v ∈ Im(P ) and let un be sequence with Pun → v. Then ∥un − um∥ ≤1c∥Pun −Pum∥→ 0, so un is a Cauchy sequence with limit u, and Pu = v, which is

why P has closed image.

Corollary 3.7.3. Let M be a closed manifold and P an elliptic differential operatorof order k. Then P ∶W k+l →W l has a finite dimensional kernel and closed image.Moreover, the dimension of the kernel of P ∶W l+k →W l does not depend on l.

Proof. The first sentence is immediate from Gardings inequality, Rellichs Lemmaand Proposition 3.7.2. The second follows from regularity.

The proof of the Fredholm property is finished by a duality consideration. Weconsider P ∶ W k+l → W l, which has closed image. To prove that the image hasfinite codimension, it is therefore enough to prove that the space of all ` ∈∈ (W l)′with ` P = 0 is finite dimensional. By duality, it has to be proven that

K = v ∈W −l∣⟨Pu, v⟩ = 0∀u ∈W k+lis finite dimensional. But

⟨Pu, v⟩ = ⟨u,P ∗v⟩and so K = ker(P ∗ ∶ W −l → W −k−l) which is finite dimensional by Corollary

3.7.3. Note that we used at this point that P ∗ is elliptic if P is elliptic. This proves

2We say that a sequence is subconvergent if it has a convergent subsequence.

50 JOHANNES EBERT

not only that P ∶W k+l →W l is a Fredholm operator, but also that the codimensionof the image does not depend on l. We summarize.

Theorem 3.7.4. Let P ∶ Γ(M ;E0) → Γ(M,E1) be an elliptic operator of orderk on the closed manifold M . Then P ∶ W k+l → W l is a Fredholm operator, withindex not depending on l. The orthogonal complement of the image of P ∶W k → L2

(taken with respect to the L2-inner product induced by a Riemannian metric on Mand hermitian bundle metrics on the vector bundles Ei) is the kernel of P ∗. Thuswe get an orthogonal decomposition C∞ = ker(P ∗)⊕ Im(P ).

Proposition 3.7.5. Let Pt, t ∈ R be a smooth family of elliptic differential operatorson a compact manifold. Then the index of Pt does not depend on t. Moreover, ifpt is a smooth family of elliptic symbols on M , then for all operators Pi, i = 0,1,with smbk(Pi) = pi, the indices are the same.

Proof. For the first part, use Theorem 3.6.3 to show that the Fredholm operator Ptdepends continuously on t, and then use Theorem 1.2.7. For the second part, onehas to show that there exists a smooth family of differential operators Pt with thesymbol pt (and then use the first part). This follows from the arguments given forProposition 2.2.19.

This last proposition is a smoking gun: it proves that the index of an elliptic dif-ferential operator only depends on the homotopy class of the principal symbol, wherehomotopy is to be understood through elliptic symbols. A suitable generalizationof this will be one of the keys for the proof of the index theorem.

The last thing we want to get out of the analysis is the Hodge decompositiontheorem. Let E = (E∗, P ) be an elliptic complex on a smooth closed manifold M .Because Pi Pi−1 = 0, we can form the cohomology of the elliptic complex:

Hp(E) ∶=ker(Pp ∶ Γ(M,Ep)→ Γ(M,Ep+1))

Im(Pp−1 ∶ Γ(M,Ep−1)→ Γ(M,Ep)).

For example, if E is the de Rham complex, then Hp(E) agrees with the usual deRham cohomology.

Equip M and the bundles with metrics, so that we can talk about the operatorD = P + P ∗, which is elliptic. Let ∆ ∶= D2 and observe that ∆ maps Γ(M,Ep)into itself. We let Hp(E) be the kernel of ∆ ∶ Γ(M,Ep) → Γ(M ;Ep). For ellipticcomplexes, the elliptic regularity theorem has the following formulation.

Theorem 3.7.6. (The Hodge decomposition theorem) Let M be a closed Riemannmanifold and E an elliptic complex on M . Then there are othogonal decompositions:

(1) Γ(M,E) = ker(D)⊕ Im(D). The kernel ker(D) is finite-dimensional.(2) ker(D) = ker(∆) = ker(P ) ∩ ker(P ∗).(3) Im(∆) = Im(D) = Im(PP ∗)⊕ Im(P ∗P ) = Im(P )⊕ Im(P ∗).(4) ker(P ) = Im(P )⊕ ker(∆).(5) The natural map ker(∆)→H(E) is an isomorphism.

Proof. (1) This is clear from Theorem 3.7.4.(2) The first equation is clear, and so is the ⊃-inclusion of the second. Con-

versely, if Pu + P ∗u = 0, then calculate 0 = ⟨Pu + P ∗u,Pu + P ∗u⟩ =⟨Pu,Pu⟩ + ⟨P ∗u,P ∗u⟩.


(3) Im(∆) = Im(D) is clear by now. Because ∆ = PP ∗ + P ∗P , it followsthat Im(D) ⊂ Im(PP ∗) ⊕ Im(P ∗P ) ⊂ Im(P ) ⊕ Im(P ∗), the orthogonalityof all spaces is clear. To prove that Pu + P ∗v ∈ Im(∆), we prove that(Pu + P ∗v)ker(∆) and use part (1). But if w ∈ ker(∆), then ⟨Pu +P ∗v,w⟩ = ⟨u + v,Dw⟩ = 0.

(4) The ⊃-inclusion is clear. If Pu = 0, then write u = x+Py +P ∗z, x ∈ ker(∆),according to parts (1) and (3). Since Pu = 0, we find that PP ∗z = 0. Thus0 = ⟨z,PP ∗z⟩ = ⟨P ∗z,P ∗z⟩, therefore P ∗z = 0 and u = x + Py.

(5) This is clear from part (4)

Remark 3.7.7. The elements of ker ∆ are called harmonic; this terminology comesfrom the de Rham complex. What is usually called Hodge theorem is the last part.It says that each cohomology class of an elliptic complex over a closed manifold hasa unique harmonic representative.

3.8. The spectral decomposition of an elliptic operator. An important prop-erty of self-adjoint elliptic operators on closed manifolds is that they admit a spec-tral decomposition. Here is our goal.

Theorem 3.8.1. Let M be a closed Riemann manifold, E → M be a hermitianvector bundle and D ∶ Γ(M,E)→ Γ(M,E) be a formally self-adjoint elliptic differ-ential operator of order k ≥ 1. For λ ∈ C, let Vλ ∶= ker(D − λ) ⊂ L2(M,E) be theeigenspace of D to the eigenvalue λ. Then the following statements hold

(1) If λ ∈ C ∖R, then Vλ = 0.(2) If λ ≠ µ, then VµVλ.(3) For each Λ ∈ R, the sum UΛ =⊕∣λ∣≤Λ Vλ is finite-dimensional and consist of

smooth sections.(4) The direct sum ⊕λ Vλ is dense in L2(M,E).

In particular, it follows from part (3) that the eigenvalues for a discrete subsetof R and that each eigenvalue has finite multiplicity. The theorem is false if D hasorder zero (find a counterexample).

Proof of the easy parts of Theorem 3.8.1. The first two parts are proven exactly asthe corresponding statements for selfadjoint endomorphisms of finite-dimensionalHilbert spaces. Namely if ∥x∥ = 1 and Fx = λx, then

λ = ⟨x,λx⟩ = ⟨x,Fx⟩ = ⟨Fx,x⟩ = λ⟨x,x⟩ = λ.If Fx = λx, Fy = µy, then

(λ − µ)⟨x, y⟩ = ⟨λx, y⟩ − ⟨x,µy⟩ = ⟨Fx, y⟩ − ⟨x,Fy⟩ = 0

so if λ ≠ µ, then ⟨x, y⟩ = 0.The third part depends on elliptic regularity. Let x ∈ UΛ. Since the operator D−λ

is elliptic, all eigenfunctions and hence x are smooth. We can write x = ∑∣λ∣≤Λ xλ,with xλ ∈ Vλ. Note that this is an orthogonal sum. Therefore

∥x∥2k = ∑

∣λ∣≤Λ

∥xλ∥2k ≤ ∑

∣λ∣≤Λ

C(∥xλ∥20 + ∥Dxλ∥2

0) ≤

≤ C(1 +Λ2) ∑∣λ∣≤Λ

∥xλ∥20 = C(1 +Λ2)∥x∥2

0.

52 JOHANNES EBERT

By Rellich’s theorem, we conclude that the ∥∥0-unit ball in UΛ is relatively com-pact, and hence UΛ is finite-dimensional.

For the last part of the proof, we recall the spectral theorem for self-adjointcompact operators in an abtract setting. Recall that the spectrum of a boundedoperator F is the set of all λ ∈ C such that F −λ is not invertible. The spectrum is acompact, nonempty subset of C, [38], Satz 23.5. Any eigenvalue is in the spectrum,but the converse does not need to hold.

Theorem 3.8.2. Let H be a Hilbert space and F ∶H →H be a compact selfadjointoperator. Then the spectrum of F is a subset of R and has 0 as its only accumulationpoint. Any spectral value of F different from 0 is an eigenvalue. The eigenspacesHλ are finite-dimensional unless λ = 0. The direct sum

ker(F )⊕⊕λ≠0

Hλ

is dense in H.

The proof is not very difficult, but would lead us too far away. See [38], Satz26.5 and Satz 26.3.

Proof of the fourth part of Theorem 3.8.1. We look at the operator L = 1 + D2,which is self-adjoint, elliptic and has order 2k. If Lu = 0, then 0 = ⟨u,u⟩+ ⟨Du,Du⟩;thus L is injective, and by Theorem 3.7.4, L ∶W 2k → L2 is invertible. Let S ∶ L2 →W 2k be the inverse; note that since L ∶ Γ(M,E) → Γ(M,E) is bijective, S maps

smooth sections to smooth sections. Let T ∶ L2 S→ W 2k → L2 be the composition,which is a compact operator by Rellich’s theorem (and the open mapping theorem).We claim that T is self-adjoint. It is continuous, and Γ(M,E) ⊂ L2 is dense, so itis enough to show that ⟨u,Tv⟩ = ⟨Tu, v⟩ holds for smooth sections. However,

⟨Lx,TLy⟩ = ⟨Lx, y⟩ = ⟨x,Ly⟩ = ⟨TLx,Ly⟩and L is surjective onto Γ(M,E), so T is self-adjoint. Moreover

⟨Lx,TLx⟩ = ⟨Lx,x⟩ = ⟨x,Lx⟩ ≥ 0

shows that ⟨Tu,u⟩ ≥ 0, i.e. that T is positive. Hence all eigenvalues of T arenonnegative. As T ∶ L2 →W 2k is bijective, 0 is not an eigenvalue of T . Therefore,by the spectral theorem for compact operators, the sum

⊕µ>0

ker(T − µ)

lies dense in L2.Consider an eigenvector Tx = µx. Since ⟨Tx,Lu⟩ = ⟨x,u⟩, we find that

⟨x, (1 − µL)u⟩ = ⟨x,u⟩ − µ⟨x,Lu⟩ = ⟨x,u⟩ − ⟨Tx,Lu⟩ = 0;

in other words, x is orthogonal to the image of (1 − µL). As µ ≠ 0, (1 − µL) iselliptic and so x is smooth, this means that all eigenfunctions of T are smooth. AsT is the inverse to L, it follows that ker(T − µ) is the 1

µ-eigenspace of L. Since D

commutes with L, the space Wµ ∶= ker(L − 1µ) is D-invariant. The operator D∣Wµ

satisfies the equation (D∣Wµ)2 + 1 = 1µ

. Thus, Wµ decomposes into the eigenspaces


of D to the two eigenvalues ±√

1µ− 1 (note that ∥T ∥ ≤ 1, whence all µ are in (0,1]).

This finishes the proof.

3.9. Guide to the literature. The local regularity theorem is a very classicalresult and is - in version or another - covered in each introductory textbook onpartial differential equations. I tried to combine arguments from different sourcesto achieve a ”best-of”. Later, we need some more analysis for the proof of the indextheorem, and I designed this chapter so that the later arguments are supported bythis approach.

Some sources avoid the Fourier transform in the definition of the Sobolev spaces,and use the norm given in Lemma 3.2.4 (which is perfectly possible, if one only usesW k for natural numbers). Instead of using Rn as the model space, one could alsotake the torus Tn and patch pieces of the torus into the manifold. This approachreplaces the Fourier transform by the (simpler) Fourier series. Fourier series areeasier because the Pontrjagin dual of Tn is the discrete group Zn. This approach ispursued in several sources, and after initial changes, the overall argument is more orless isomorphic to the one using the Fourier transform. Examples for this approacharethe books by Warner [68], Griffiths-Harris [31] and Higson and Roe [36], [56]

I think that the role of the symbol is more cleanly reflected by the self-dualityof Rn. Also, the proof of the most general version of the index theorem (in theoriginal paper by Atiyah and Singer [8] requires a more general class of operators,namely pseudodifferential operators, and one needs the Fourier transform to definethem. Once the formalism of pseudodifferential operators is set up, the proof of theregularity theorem can be streamlined considerably [70]. I have written this part ofthe notes in order to make contact with the theory of pseudodifferential operatorsmore easily possible.

The treatment of the Fourier transform is taken from Lang [46]. The basictheorems on Sobolev spaces (Sobolev embedding, Rellich, duality) are proven in[48], [27], [70], [15], with essentially the same argument that we gave. The proofof Gardings inequality is that from [68], with the changes needed to suit into ourframework. In [70], [48], [27], Theorem 3.3.2 is derived using the calculus of pseu-dodifferential operators. In [31], [36], [56], the special structure of the operatorsstudied is used heavily.

Friedrichs lemma is an ”exercise” in [56] and a structured exercise in [65]. Theproof we gave follows the outline in [65]. Sources as [68] and most PDE textbooks Ilooked into ([24], [65], [63]) replace the use of the mollfiers by a different smoothingprocedure, which might be technically simpler. One can also use distribution theory.

The globalization of the Sobolev theory is a standard exercise (and done in mostof the above sources). I have no specific source for the proof of the Fredholmproperty and the spectral secomposition.

54 JOHANNES EBERT

4. Some interesting examples of differential operators

4.1. The Euler number. Let Mn be a closed smooth manifold of dimension n.Recall the de Rham complex A∗(M), which is an elliptic complex. To get anelliptic operator out of the de Rham complex, we need a Riemann metric on Mand a hermitian bundle metric on the vector bundle ΛpT ∗M . There is a canonicalchoice of such a hermitian metric, depending on the Riemann metric on M .

Lemma 4.1.1. Let V be an n-dimensional euclidean vector space, with an or-thonormal basis (e1, . . . , en). For I ⊂ n, ∣I ∣ = p, consider the basis element eI ∈ΛpV ∗. Define a hermitian metric on ΛpV ∗ by declaring (eI)I⊂n;∣I ∣=p to be an or-thonormal basis. This hermitian inner product does not depend on the choice of theorthonormal basis.

Proof. We prove the following equivalent statement. Let V = Rn and use thestandard basis to define the inner product on ΛpRn. We claim that the actionof the group O(n) on ΛpRn is via isometries. The next lemma gives a system ofgenerators of O(n) and it is easy to check that the generators act by isometries.

Lemma 4.1.2. Let Gn ⊂ O(n) be the subgroup that is generated by the permutationmatrices and the matrices of the form

⎛⎜⎝

cos(t) − sin(t)sin(t) cos(t)

1

⎞⎟⎠

(t ∈ R). Then Gn = O(n).

Proof. This can be seen by induction on n. The case n = 2 is easy. For the inductionstep one uses the elementary fact that if a group G acts transitively on a set X andH ⊂ G is a subgroup that acts transitively such that for a fixed x ∈ X the isotropygroups Hx and Gx are equal, then H = G. By induction on n, one proves thatGn acts transitively on Sn−1. The isotropy group of O(n) at en is O(n − 1), and(Gn)en ⊃ Gn−1. By induction hypothesis, Gn−1 = O(n − 1).

If Mn is a closed RiemIf Mn is a closed Riemann manifold, we thus get a canonical bundle metric on

Λ∗T ∗M . The elliptic operator associated with the de Rham complex is the operator

D = d + d∗ ∶ Aev(M)→ Aodd(M).The Hodge decomposition theorem can be used to compute the index of D.

Theorem 4.1.3. Let M be a closed manifold. The index of D ∶ Aev(M) →Aodd(M) is the Euler characteristic χ(M) ∶= ∑np=0(−1)p dimHp(M) of M .

Proof. Let Hp(M) ⊂ Ap(M) be the space of harmonic forms on M . By the Hodgetheorem, the natural map Hp(M)→Hp(M) is an isomorphism. Moreover,

ind(D) =n

∑p=0

(−1)p dimHp(M) = χ(M).


4.2. The signature. So far, we have completely computed the index of one dif-ferential operator that exists on any manifold M , namely the operator d + d∗. Ifthis were the only interesting operator, there would be no ”index theory”. It turnsout that we need more structure on a manifold to get new operators linked tothat extra structure. The first such extra structure is an orientation. But letus go back to the operator D = d + d∗ ∶ A∗(M) → A∗(M) for a second. Ob-serve that D = d + d∗ ∶ A∗(M) → A∗(M) is formally self-adjoint; therefore itsindex is zero and the operator itself is not very interesting from the perspectiveof index theory. In connection with the Euler characteristic, we studied the de-composition A∗(M) = Aev(M)⊕Aodd(M) and we obtained an interesting operatorD0 ∶ Aev(M) → Aodd (by restricting D). Let us formulate this a bit differently.Let I ∶ A∗(M) → A∗(M) be the operator that is (−1)p on Ap(M). This is aninvolution, which is self-adjoint and comes from an involution of the vector bundleΛ∗T ∗M (important!). The spaces Aev/Aodd are the +1/ − 1-eigenspaces of I. Theimportant fact that we used secretly is that

DI = −ID,(both anticommute). If we decompose A∗(M) according to the eigenspaces of

I, we get

I = (1−1

) ; D = (D2 D1

D0 D3) ; D∗ =D.

The equation DI + ID = 0 means (quick computation) that D2 = D3 = 0 andD1 =D∗

0 , i.e.

D = ( D∗0

D0)

and

ker(D) = ker(D0)⊕ ker(D∗0) = ker(D0)⊕ Im(D0).

The involution I maps ker(D) to itself (if Dx = 0, then DIx = −IDx = 0) andwe get the equalities

ind(D0) = dim(ker(D0)) − dim(ker(D1)) = Tr(I ∣ker(D)).If we take I as above, we get the Euler number of M . We refer to I as a grading

of the de Rham complex. We solidify these observations in a definition.

Definition 4.2.1. Let M be a closed Riemannian manifold, E →M be a hermit-ian vector bundle and D ∶ Γ(M,E) → Γ(M,E) be a formally self-adjoint ellipticdifferential operator. A grading of D is an orthogonal involutive vector bundle iso-morphism ι ∶ E → E such that Dι = −ιD, with eigenbundles E±. With respect tothe decomposition E = E+ ⊕E−, write

D = ( D−D+

) .

The index of (D, ι) is

ind(D, ι) = ind(D+) = − ind(D−) = Tr(ι∣ker(D)) ∈ Z.

56 JOHANNES EBERT

Such a pair (D, ι) is called a graded selfadjoint elliptic operator.

If P ∶ Γ(E0) → Γ(E1) is an arbitrary elliptic operator, we get a graded self-adjoint one, by setting E = E0 ⊕ E1 (orthogonal sum), ι = (−1)i on Ei and theoperator is

( P ∗

P)

Thus ordinary elliptic operators and graded self-adjoint ones are essentially thesame thing. The point is that the grading is often easier to describe than theeigenspaces! One can change the grading by a sign, the index changes by sign aswell: ind(D,−ι) = − ind(D, ι).

But for the de Rham complex on an oriented manifold, there is a more substantialchange of the grading.

Lemma-Definition 4.2.2. Let V be an n-dimensional oriented euclidean vectorspace. The star operator is the uniquely determined operator ⋆ ∶ ΛpV ∗ → Λn−pV ∗

such that the identity

⟨ω, η⟩vol = ω ∧ ⋆ηholds for all forms ω,β. For ω ∈ ΛpV ∗, one has

⋆ ⋆ ω = (−1)n(n−p)ω.

Theorem 4.2.3. Let Mn be an oriented Riemann manifold. Then the adjointd∗ ∶ Ap(M)→ Ap−1(M) is given by

d∗ω = (−1)pn+n+1 ⋆ d ⋆ ω.

This is an easy (but tedious) consequence of the Stokes theorem, but has aprofound consequence. We restrict to real-valued forms. Let us denote by

Hp(M) = ker(∆ ∶ Ap(M)→ Ap(M))the space of harmonic forms. By the Hodge decomposition theorem, we know

that

Hp(M) ≅Hp(M ;R)

Proposition 4.2.4. If M is an oriented Riemann manifold and ω a harmonicform on M , then ⋆ω is harmonic.

Thus ⋆ induces an isomorphism ⋆ ∶ Hp(M) → Hn−p(M), and thus by the iso-morphism Hp(M) ≅Hp(M) an isomorphism

Hp(M) ≅Hn−p(M)which depends on the Riemann metric. Thus we get a proof of a weak form of

Poincare duality. Let us remark that we killed a fly with a sledgehammer: there isa simple proof of Poincare duality in the framework of de Rham cohomology, see[17]. Let us have a slightly closer look.

Theorem 4.2.5. Let M be a closed oriented manifold. Then the pairing Hp(M)⊗Hn−p(M)→ C, α⊗ β ↦ ∫M α ∧ β is a perfect pairing.


Proof. By the Hodge theorem, it suffices to consider the pairing on H∗. Compute,for ω ∈Hp and η ∈Hn−p:

∫Mω ∧ η = (−1)p(n−p) ∫

Mω ∧ ⋆ ⋆ η = (−1)p(n−p) ∫

M⟨ω;⋆η⟩vol = ⟨ω;⋆η⟩.

and since ⋆ is an isomorphism, this is clearly a nondegenerate pairing (and perfectbecause all spaces involved are finite-dimensional).

On even-dimensional manifolds, we get a finer structure. Recall that the spacesAp(M) of complex-valued forms come with a natural real structure (i.e. a conjuga-tion map) and that the operators ⋆, d and d∗ are all real operators (commute withthe conjugation). There are two cases of even dimensions: n = 4k + 2 (moderatelyinteresting) and n = 4k (very interesting). Let us, in both cases, restrict to themiddle dimension. Let n = 2m. Note that on Am(M2m), one has

⋆2 = (−1)m.On Hm(M ;R), we have two bilinear forms. One is given by

Ω ∶ (ω, η)↦ ∫Mω ∧ η,

and this has a purely homological meaning (no metric is used to define it). Andthere is the scalar product

⟨ω;η⟩ ∶ ∫Mω ∧ ∗η

which is defined on Hm(M) (and thus uses the metric). The two forms arerelated by

⟨ω;η⟩ = Ω(ω,⋆η).In the case of odd m, the form Ω is symplectic (i.e. skew-symmetric and nonde-

generate). On the other hand, ⋆2 = −1 and thus it defines a complex structure onHm(M). If you are familiar with the terminology of symplectic linear algebra, thenΩ is a symplectic form on Hm(M), ⋆ is a compatible complex structure (dependingon the metric on M).

The case n = 4k is extremely interesting. Consider the symmetric, nondegeneratebilinear form Ω on H = H2k(M). There exists an orthogonal basis of H (i.e.Ω(ei, ej) = εiδij , εi = ±1). The Sylvester inertia theorem from linear algebra saysthat the number sign(Ω) ∶= ♯i∣εi = 1 − ♯i∣εi = 1− does not depend on the choiceof the basis.

Definition 4.2.6. Let M4k be an oriented closed manifold. The signature of M isthe signature of the bilinear form Ω on H2m(M).

This is a fundamental invariant in differential topology. One of its meanings isthat the signature is a bordism invariant; if W 4k+1 is an oriented compact manifoldwith boundary M , then sign(M) = 0. The signature plays an important role in theclassification theory of high-dimensional manifolds.

Problem 4.2.7. Express the signature in terms of characteristic classes.

58 JOHANNES EBERT

This problem was solved by Hirzebruch in 1954, using topological methods de-veloped by Thom. The Hirzebruch signature formula was one of the motivatingexamples for the search of the general index formula, and in this course, we willprove the signature formula as a special case of the index theorem. We have not yetstated the signature formula (the right-hand-side would not yet be understandable).But we can go the first step along its proof, and this is by identifying the signatureas the index of a new elliptic operator (which exists only on oriented manifolds ofdimension 4k).

Lemma 4.2.8. Let M4k be oriented and closed. Then sign(M) is equal to the traceTr(⋆∣H2n(M)).

Proof. Since on 2k-forms, ⋆2 = 1, we see that Ω(ω, η) = ⟨ω;⋆η⟩. The rest of theproof is pure linear algebra. Let V be a finite-dimensional vector space, I aninvolution, B a symmetric bilinear form and ⟨; ⟩ and assume these are related by⟨x; y⟩ = B(x, Iy). Since

B(x, Iy) = ⟨x; y⟩ = ⟨y;x⟩ = B(y, Ix) = B(Ix, y),the involution I is selfadjoint, and the decomposition V = V+ ⊕ V− into the

eigenspaces of I is orthogonal. If x ∈ V±, we get B(x,x) = ⟨x; Ix⟩ = ±⟨x;x⟩, and ±Bis positive definite on V±. If x ∈ V+ and y ∈ V−, then B(x, y) = ⟨x; Iy⟩ = −⟨x; y⟩ =0 because both spaces are orthogonal. Thus the signature is dimV+ − dimV− =Tr(I).

4.3. Complex manifolds and vector bundles. We now investigate the refine-ment of the harmonic theory for complex manifolds. We first begin in arbitrarydimensions; but at a crucial point it turns out that one dimensional complex man-ifolds are much easier to treat. The index theorem on Riemann surfaces is a veryclassical result: the Riemann-Roch formula.

Definition 4.3.1. Let M be a smooth manifold of dimension 2n. A smooth at-las (Ui, hi) is holomorphic if all transition functions are holomorphic. A complexstructure is a maximal holomorphic atlas, and a complex manifold is a manifoldM , together with a complex structure. A 1-dimensional complex manifold is calledRiemann surface.

Examples: C, CP1, tori. Moreover, one can prove that each differentiable surfaceof genus g has a complex structure (by no means unique, and this is by no meansan easy result).

Definition 4.3.2. Let M be a complex manifold and V →M be a complex vectorbundle. A bundle atlas (Ui, hi) of V is holomorphic if the transition functionshij ∶ Uij → GLr(C) are holomorphic.

Examples 4.3.3. The tautological line bundle on CPn is a holomorphic vector bun-dle. The tangent bundle of a complex manifold has a natural holomorphic structure.On the dual vector bundle E∗, there is a holomorphic structure. Likewise, tensorproducts and hom-bundles have holomorphic structures.

We discuss the tangent bundle to a complex manifold in a little more detail. Let

x ∈M and let x ∈ U h→ h(U) ⊂ Cn be a holomorphic chart. The composition


Jx ∶ TxMTh→ Th(x)Cn

can≅ Cn i⋅→ Cn

can≅ Th(x)

Th−1→ TxM

satisfies J2x = −1, does not depend on the choice of h (since the atlas is holomor-

phic). J is a smooth bundle endomorphism and satisfies J2 = −1. This turns TM(which is a priori only a real vector bundle) into a complex vector bundle.

Definition 4.3.4. Let M be a real manifold. An almost-complex structure onM is a complex structure on the vector bundle TM , in other words, a smoothendomorphism J of TM with J2 = −1.

One can prove that on surfaces, each almost complex structure is induced froma complex structure. This requires an amount of analysis (not directly related toindex theory). The corresponding fact in higher dimensions is false (but there is anadditional condition on J that guarantees this).

4.4. Multilinear algebra of complex vector spaces. We now have to delveinto (multi)linear algebra of complex vector spaces. Let V be a real vector spaceof dimension 2n, equipped with a complex structure J . This defines the structureof a complex vector space on V , namely (a + ib)v ∶= av + bJv. We consider Λ∗V ∗,the algebra of complex-valued, R-multilinear alternating forms. This is a complexvector space, the piece ΛpV ∗ has complex dimension (2n

p). There is a conjugation

map ω ↦ ω on Λ∗V ∗, defined by

ω(v1, . . . , vp) ∶= ω(v1, . . . , vp).If e1, . . . , en is a C-basis of V and e1, . . . , en the dual basis of V ∗ = Λ1V ∗, then

(e1, e1, . . . , en, en) is an R-basis. Then the set

ei1 ∧ eip ∧ ej1 ∧ ejq ∣p + q = r, i1 < . . . < ip; j1 < . . . < jqis a C-basis of ΛrV ∗. We define subspaces

Λp,qV ∗ ∶= spanei1 ∧ eip ∧ ej1 ∧ ωejq ∣i1 < . . . < ip; j1 < . . . < jq.It is clear that ⊕p+q=r Λp,qV ∗ = ΛrV ∗ and that dim(Λp,qV ∗) = (n

p)(nq) and

Λp,qV ∗ = Λq,pV ∗. In basis-free terms, Λp,qV ∗ is the subspace of all ω ∈ Λp+qV ∗

such that for all v1, . . . , vr ∈ V and all z ∈ C×, one has

ω(zv1, . . . , zvr) = zpzqω(v1, . . . , vr).Complex vector spaces are naturally oriented; if (e1, . . . , en) is a C-basis, then

the R-basis (e1, ie1, . . . , en, ien) is said to be positively oriented.If V is a complex vector space, a compatible scalar product is a R-valued scalar

product ⟨; ⟩ such that J is an orthogonal map. A compatible scalar product extendsto a complex scalar product by

h(v,w) = ⟨v;w⟩ − i⟨v;Jw⟩.However, when V is the tangent space to a complex manifold, we use only R-

valued scalar products. If ⟨; ⟩ is a compatible scalar product, one can find anorthonormal basis of the form (e1, ie1, . . . , en, ien). This is seen by taking a complexorthonormal basis with respect to h.

60 JOHANNES EBERT

The Hodge star operator is most usefully not extended as a C-linear operator,but as a C-antilinear operator ⋆ (the usual Hodge star, followed by complex con-jugation). The extension of the inner product on the real valued forms to complexvalued forms is given by the formula

⟨ω;η⟩vol = ω ∧ ⋆η.The volume form is an element of Λn,nV ∗.

Lemma 4.4.1.

(1) The spaces Λp,q for different values of (p, q) are orthogonal.(2) The Hodge operator ⋆ takes Λp,qV ∗ to Λn−p,n−qV ∗.

Proof. The group S1 ⊂ C× acts by isometries on V , since the metric is compatible.Thus the induced action on Λ∗V ∗ is by isometries. The subspace Λp,qV ∗ is thesubspace on which S1 acts by the character z ↦ zp−q. Thus the spaces Λp,qV ∗ areorthogonal (by the same argument that proves that the eigenspaces of a unitarymatrix are orthogonal). The second statement follows by the formula for the scalarproduct.

The next lemma, easy as it is, is nothing short of a miracle, it allows us todescribe the interplay between compatible metrics and the complex structure forRiemann surfaces. In higher dimensions, the situation is much more complicated.

Lemma 4.4.2. Let V be a 1-dimensional complex vector space with a compatiblemetric. For all ω ∈ Λ1V ∗, the identity

ω J = − ⋆ ωholds.

Proof. A straightforward check on a basis: Let (e1, e2 = Je1) be an oriented R-basisof V . Then

e1 J(e1) = 0; e1 J(e2) − 1; e2 J(e1) = 1; e2 J(e2) = 0

and

⋆e1(e1) = e2(e1) = 0; ⋆e1(e2) = e2(e2) = 1; ⋆e2(e1) = −e1(e1) = −1; ⋆e2(e2) = −e1(e2) = 0.

These notions generalize to forms with values in a fixed (finite-dimensional)hermitian vector space (E,h). The natural map τ ∶ E → E∗, e ↦ h(e, ) is aC-antilinear isomorphism. We define

⋆E ∶ Λp,qV ∗ ⊗E → Λn−p,n−qV ∗ ⊗E∗

by

ω ⊗ e↦ ⋆ω ⊗ τ(e),an antilinear isometry.


4.5. The Dolbeault (Cauchy-Riemann) operator. Let M be a complex man-ifold. From the linear algebra in the previous section, we get a decompositionAr(M) = ⊕p+q=rAp,q(M), Ap,q(M) ∶= Γ(M,Λp,qT ∗M). In local holomorphiccharts, we can write forms in Ap,q(M) as sums of forms of the form

adzi1 ∧ . . . dzip ∧ dzj1 ∧ dzjq ; a ∈ C∞.

If we define, in local coordinates, the operators

∂

∂zj∶= 1

2( ∂

∂xj− i ∂∂yj

); ∂

∂zj∶= 1

2( ∂

∂xj+ i ∂∂yj

),

we get that for functions a:

da =n

∑j=1

∂

∂zjadzj +

∂

∂zjadzj .

This implies that

dAp,q(M) ⊂ Ap,q+1(M)⊕Ap+1,q(M)and we set, for ω ∈ Ap,q(M); dω = ∂ω + ∂ω, with ∂ω ∈ Ap+1,q and ∂ω ∈ Ap,q+1.

Remark 4.5.1. The holomorphic functions are the solutions of ∂f = 0. Moreover∂2 = ∂2 = 0 and ∂∂ + ∂∂ = 0.

Lemma 4.5.2. Let E →M be a holomorphic vector bundle and let Ap,q(M,E) bethe space of (p, q)-forms with values in E. Let s be a local holomorphic section ofE. If ω ∈ Ap,q(M), we define ∂E(ω ⊗ s) ∶= (∂ω)⊗ s. Then

(1) ∂E is a well-defined differential operator Ap,q(M,E) → Ap,q+1(M,E) oforder 1.

(2) The symbol is symb∂E(ξ)e = iξ0,1 ∧ e, where ξ ∈ Λ0,1T ∗M denotes the pro-

jection of ξ.

(3) 0→ A0,0(M,E) ∂→ A0,1(M,E) ∂→ . . .A0,n(M,E)→ 0 is an elliptic complex.(4) Λp,0(T ∗M)→M is a holomorphic vector bundle, and the diagram

Ap,q(M) ∂ //

≅

Ap,q+1(M)

≅

A0,q(M ; Λp,0T ∗M) ∂ // A0,q+1(M ; Λp,0T ∗M)commutes.

Proof. (1) follows because we used a holomorphic local section. (4) is easily seenin local coordinates. The symbol is computed as follows. Let f be a function withdxf = ξ and s a local holomorphic section with s(x) = e and ω a (p, q)-form. Then

symb∂(ξ)(ω ⊗ e) = i[∂E , f](ω ⊗ s)(x) = i∂E(fω ⊗ s) − if ∂E(ω ⊗ s) = i∂f ∧ ω ⊗ sby the Leibniz rule for the ∂-operator. But ∂f(x) = ξ0,1. For (3), it is clear that

∂2E = 0. The exactness of the symbol sequence is proven exactly as in the real case;

the additional argument needed is that ξ ↦ ξ0,1 is an isomorphism T ∗RM → Λ0,1T ∗M(dxi ↦ 1/2dzi, dyi ↦ i/2dzi).

62 JOHANNES EBERT

Note that there is no canonical possibility to extend the operators ∂ to vectorvalued forms (a suitable connection on E will give such a possibility).

Problem 4.5.3. (The Riemann-Roch problem) Compute the index of the Dol-beault complex for a holomorphic vector bundle E →M on a complex closed man-ifold.

We undertake the first steps in this lecture. Then we specialize to the case ofdimension 1. This is the classical Riemann-Roch problem, as we will see. Then wetake the first steps towards the solution of the index problem on a Riemann surface.This will motivate the introduction of two major players: characteristic classes andK-theory. In higher dimensions, the index formula for the Dolbeault complex goesunder the name Hirzebruch-Riemann-Roch theorem, and we will prove this as aspecial case of the general index formula.

We now assume that our complex manifold M comes with a compatible Riemannmetric.

Proposition 4.5.4. The adjoint of ∂Ap,q(M)→ Ap,q+1(M) is given by ∂∗ = −⋆∂⋆.More generally, the adjoint of ∂E is ∂E∗ = −⋆∂E ⋆. (See [70] p. 168).

If E → M is a holomorphic vector bundle, we let Hp(M,E) be the p th coho-mology of the elliptic complex ∂E .

Theorem 4.5.5. (Serre duality) There is a conjugate linear isomorphism Hp(M,E) ≅Hn−p(M,Λn,0T ∗M ⊗E∗).

Proof. There is a diagram that commutes

A0,p−1(M,E)

⋆

A0,p(M,E)∂∗E

oo ∂E //

−⋆

A0,p+1(M,E)

⋆

An,n−p+1(M,E∗) An,n−p(M,E)∂E∗

oo∂∗E∗ // An,n−p−1(M,E∗).

The respective cohomology groups are, by the Hodge theorem Hp(M,E) =ker(∆E), equal to the intersection of the kernels of the horizontal maps. Theresult follows (note that the vertical arrows are antilinear).

4.6. The Hodge decomposition on a Riemann surface. Let M be a closedconnected Riemann surface.

Definition 4.6.1. The genus of M is the number g ∶= 12

dimH1(M,R).The genus is an integer by Poincare duality. We study the de Rham complex of

M :

A0,1

∂

""A0,0

∂

<<

∂

""

A1,1

A1,0

∂

<<

and define Hp,q =Hp+q∩Ap,q, the space of d-harmonic p, q-forms. Since d is real,we have ⋆d⋆ = ⋆d⋆.


Theorem 4.6.2. For a closed Riemann surface, the following hold:

(1) Hp,q =Hq,p.(2) Hr =⊕p+q=rHp,q.(3) H0,0 = ker(∂).(4) H1,0 = ker(∂)(5) H0,1 = ker(∂∗) = Im(∂).(6) H1,1 = ker(∂∗) = Im(∂).

Proof. (1) is clear. (2) is easy if r ≠ 1 (in these cases, there is only one summand).Let ω ∈ H1 be harmonic. Then ⋆ω is harmonic and thus ω J as well, by Lemma4.4.2. But A1,0 and A0,1 are the ±i-eigenspaces of J , and the projection onto theseis therefore still harmonic. (2) follows.

For the other four part, we first consider the easy inclusions.(3i) ω ∈H0,0 ⇒ 0 = dω = ∂ω + ∂ω, and both summands have to be zero.(4i) If ω ∈H1,0, then dω = 0 = ∂ω + ∂ω = ∂ω (∂ω ∈ A2,0 = 0).(5i) ω ∈ H0,1 ⇒ 0 = d∗ω = ∂∗ω + ∂∗ω = ∂∗ω for degree reasons. The second

equality follows from the main regularity theorem.(6i) ω ∈ H1,1 ⇒ 0 = d∗ω = ∂∗ω + ∂∗ω = ∂∗ω for degree reasons. The second

equality follows from the main regularity theorem.(4ii) ∂ω = 0. Then, for degree reasons, ∂ω = 0 and ω is closed. On the other

hand, d∗ω = −⋆d⋆ω = −⋆d ∗ ω. As ω ∈ A0,1, ⋆ω = −ω J = iω. Therefore −⋆d ⋆ ω =−⋆diω = i⋆dω = 0, and so ω is harmonic.

(3ii) ∂f = 0, then 0 = ddf = ∂∂f + ∂∂f = ∂∂f . Thus ∂f is exact and in ker(∂).By (4), ∂f is also harmonic, and thus harmonic and exact, therefore zero.

(5ii) 0 = ∂∗ω = −⋆∂⋆ω. Therefore, ⋆ω = ⋆ω ∈ A1,0 is ∂-closed and thereforeharmonic by (4).

(6ii) ω ∈ A1,1, ∂∗ω = 0. Then 0 = d∗d∗ω = ∂∗∂∗ω + ∂∗∂∗ω = ∂∗∂∗ω. By (5), ∂∗ωis harmonic and coexact, therefore 0.

Remark 4.6.3. Part (1) of Theorem 4.6.2 holds for all compact complex manifolds,with the same proof. Parts (3)-(6) are specific to the complex dimension 1: in higherdimensions, the individual operators ∂ are not elliptic, and you should not expectthe kernels/cokernels to be finite-dimensional. For complex manifolds of higherdimensions, one would part (2) of the above theorem to be true, i.e.

Hk(M) = ⊕p+q=k

Hp,q(M).

This is not true in general. For example, if it is true, then dimH1(M) = dimH1(M) =2 dimH1,0(M) has to be even. But S1 × S3 does not have this property, and stillhas a complex structure: let Z act on C2 ∖ 0 by n ⋅ z ∶= λnz, λ ∈ C, ∣λ∣ ≠ 0,1.This action is properly discontinuous and by biholomorphic maps, so the quotient(C2 ∖ 0)/Z ≅ S1 × S3 inherits a complex structure. These complex manifolds arecalled Hopf surfaces.

The additional condition for the decomposition to hold is that the metric onM is Kahler, meaning that the 2-fom ω(X,Y ) = g(X,JY ) is closed. The complexprojective space CPn has an essentially unique (up to multiplication by a constant)Riemann metric, the Fubini-Study metric, so that the canonical action of U(n + 1)is by isometries. This metric is obtained by ”averaging” an arbitrary compatible

64 JOHANNES EBERT

metric using the invariant integral on U(n + 1). Hence the form ω is U(n + 1)-invariant as well.

Complex submanifolds of Kahler manifolds are Kahler. This means that eachcomplex submanifold of CPn, i.e. a complex projective variety, is Kahler. Therefore,Kahler metrics play an important role in complex algebraic geometry.

The key to the Hodge decomposition for Kahler manifolds are the Kahler iden-tities relating the Laplace operator ∆ to the operators ∂ and ∂. The proof of theKahler identities is only differential calculus, but far from trivial and lies beyondthe scope of this lecture.

Suggested further reading: The classic text on complex algebraic geometry is[31]. A more terse exposition of the Hodge decomposition in the complex case isin [70]. More recent sources are [11] (more to the differential-geometric side of thestory) and [66], [67], more to algebro-geometric side.

Theorem 4.6.4. Let M be a compact connected Riemann surface. Then

(1) ∂ ∶ A0,0(M)→ A0,1(M) has index 1 − g.(2) ∂ ∶ A1,0(M)→ A1,1(M) has index g − 1.

Proof. By the previous theorem, one sees that

ind(∂Λ0,0) = dimH0,0 − dimH0,1; ind(∂Λ1,0) = dimH1,0 − dimH1,1.

But dimH1,0 + H0,1 = dimH1 = dimH1(M) = 2g by Theorem 4.6.2 (2) anddimH1,0 = dimH0,1, so both numbers equal g. A harmonic 0-form is closed, henceconstant, whence dimH0,0 = 1. Finally dimH1,1 = dimH1 = 1.

The above index computation turns out to be enough for the computation ofind(∂E) for a general holomorphic vector bundle V → M over a Riemann sur-face. We have accumulated enough knowledge to take one further step towards thegeneral case.

Proposition 4.6.5. Let M be a Riemann surface and V → M be a holomorphicvector bundle. Then the index ind(∂V ) ∈ Z depends only on the vector bundle V ,not on the holomorphic structure.

Proof. We have seen that the symbol of ∂E is symb∂E(ξ) = iξ0,1 ∧ , and thismeans that the symbol only depends on the complex structure of M , not on theholomorphic structure on E. If two holomorphic structures are given on E, denotethe Cauchy-Riemann operators by ∂0

E and ∂1E . For each t ∈ [0,1], the operator

Dt ∶ (1 − t)∂0E + t∂1

E is elliptic and has the same symbol.Thus we get a path [0,1]→ Fred(W 1(M,E);L2(M,E)), t↦Dt, and this path is

continuous. Since the index is homotopy invariant, we see that ind∂0E= ind∂1

E.

Proposition 4.6.6. Let E →M be a complex vector bundle on the Riemann sur-face. Then σ(ξ) = iξ ∧ is an elliptic symbol, and there is an elliptic operator DE

with that symbol.

This is clear (existence of differential operators with given symbol).

Definition 4.6.7. Denote by Vect(M) the set of isomorphism classes of complexvector bundles. It becomes a commutative semigroup by taking direct sums ofvector bundles.


The previous two propositions show that

[V ]↦ ind(DV )is a well-defined homomorphism map Vect(M)→ Z. It is trivial that ind(DV ⊕W ) =

ind(DV ) + ind(DW ), and we have a semigroup homomorphism.

Proposition 4.6.8. Let (A,⊕) be a commutative semigroup. Let F (A) be thequotient of the free abelian group ZA, by the subgroup generated by the elements

a⊕ b − a − band let ι ∶ A → F (A), a ↦ a, be the natural homomorphism. Then if B is any

abelian group and f ∶ A→ B a homomorphism of semigroups, then there is a uniquegroup homomorphism g ∶ F (A)→ B such that g ι = f . If A is already a group, theι is an isomorphism.

Proof. This is clear (universal property formal nonsense).

Definition 4.6.9. Let X be a compact Hausdorff space. The K-theory group ofX is K0(X) ∶= F (Vect(X)).

Let us summarize what we have achieved so far.

Proposition 4.6.10. Let M be a Riemann surface. There is a unique homomor-phism I ∶ K0(M) → Z such that for each holomorphic vector bundle V → M , theidentity I(V ) = ind(∂V ) holds.

The rest of the proof of the index theorem for Riemann surfaces will now be:

Find numbers that one can attach to complex vector bundles on a surface(one will be of course the rank, the other will be the Chern number).

Prove that the bundles C and Λ1,0 generate K0(X) in a suitable way andfind the right linear combination of the numerical invariants.

4.7. Relation to the classical theory. In the literature on Riemann surfaces,the Riemann-Roch theorem is typically not stated as an index theorem for anelliptic operator. Let us briefly describe the classical outlook of the theorem. LetX be a compact Riemann surface. A formal linear combination D = ∑ri=1 nipi,pi ∈ X points, ni ∈ Z, is called a divisor. We identify the relation 0pi = 0 andnp +mp = (m + n)p; with these conventions the set of divisors becomes an abeliangroup div(X). A divisor is nonnegative if ni ≥ 0 and we say D1 ≥ D0 if D1 −D0 isnonnegative.

For example, consider a meromorphic function f on X. Let p1, . . . , ps be thezeroes and let ni be the order of f at pi. Moreover, let ps+1, . . . , pr be the poles,with order −ni. We denote by (f) = ∑ri=1 nipi the divisor of f . More generally, if fwere a meromorphic section of a line bundle, we can apply the same idea and geta divisor (f) on X.

The degree of the divisor D is the sum ∑i ni ∈ Z. A divisor is principal if thereexists a meromorphic function f with D = (f).

To any divisor, one can construct a line bundle LD, in the following canonical way(this uses the cocycle description of vector bundles). Let D = ∑i nipi be a divisor,written in minimal form. Let U0 = X − pi, and let Ui be a disc neighborhoodof pi. Pick holomorphic charts hi ∶ Ui → E, hi(pi) = 0 and assume that the Uiare disjoint for i ≥ 1. Let LD = ∐iUi × C/ ∼; the equivalence relation is that

66 JOHANNES EBERT

U0 × C ∋ (x, z) ∼ (x, zhi(x)ni) whenever x ∈ U0 ∩ Ui. With the obvious projectionto X, this becomes a holomorphic line bundle LD.

This line bundle comes equipped with a meromorphic section; namely, takesD(x) = 1 over U0. Inspection shows that (sD) = D holds. The constructionsatisfies LD0+D1 ≅ LD0 ⊗ LD1 . If D = (f) for a meromorphic function, the bun-dle LD is trivial, because f−1sD is a meromorphic section without zeroes or poles.More generally, if s is a meromorphic section of a line bundle, then L(s) ≅ L. It fol-lows, by the Poincare-Hopf theorem, that the degree of D equals the Chern number

∫X c1(LD).Let M×(X) be the multiplicative group of nonzero meromorphic functions and

H1(X,O×) the group of isomorphism classes of holomorphic line bundles, we getan exact sequence

0→ C× →M×(X)→ div(X)→H1(X,O×)(it is indeed a cohomology sequence of a sequence of sheaves). The image of the

last map is the group of all line bundles that have a meromorphic section. Theseare all line bundles (and so the sequence is exact at the end).

Lemma 4.7.1. Each holomorphic line bundle over a Riemann surface has a nonzeromeromorphic section.

Proof. Let the Chern number c of L be at least 2g − 1. Then, by Serre dualitydim coker∂L = dim ker ∂Λ1,0⊗L∗ . By Poincare-Hopf, this is zero, since the Chernnumber of Λ1,0 ⊗ L∗ equals 2g − 2 − c < 0. By Riemann-Roch, we conclude thatdim ker ∂L = ind ∂L = 1 − g + d > 0. Thus each line bundle of large degree has ahomlomorphic section.

For a given line bundle L, pick a line bundle L′ such that c(L′)− c(L) and c(L′)are both at least 2g − 1. We get a holomorphic section s of L′ and t of L′ ⊗ L∗.Then st−1 is the desired meromorphic section.

Now suppose that s is a holomorpic section of LD. In the chart over U0, we getsimply a holomorphic function g. At a point pi with ni < 0, g must have a zero, oforder at least −ni, while if ni > 0, g has at worst a pole of order ni.

So we see:

Lemma 4.7.2. The space ker(∂LD) of holomorphic sections of LD is isomorphicto the space of meromorphic functions g such that (g) ≥D.

The bundle Λ1,0 is called the canonical bundle in the classical theory. Any divisorassociated with a meromorphic section of K is called canonical divisor and denotedK. Using Serre duality, we might now restate the Riemann-Roch theorem:

Theorem 4.7.3. (Riemann-Roch, classical version) For each divisor D on a com-pact Riemann surface of genus g, of degree d, we have

dim ker(∂LD) − dim ker(∂LK−D) = 1 − g + d.

It is worth to work out everything for CP1. For the proof of the Riemann-Rochtheorem, we also need to look at the case g = 1.

Proposition 4.7.4. Let X be a Riemann surface of genus g = 1 and let x ∈ X.Then the index of ∂L(x) is equal to 1.


Proof. First we show that Λ1,0 is the trivial holomorphic line bundle. This isobvious if we use the fact that X must be a complex torus C/Γ, but we do notwish to rely on that. Instead, by Theorem 4.6.2, the space of holomorphic sectionsof Λ1,0 is one-dimensional. Pick a nonzero holomorphic section ω. As the Chernnumber of Λ1,0 is zero, by the topological Gauss-Bonnet theorem, and because alllocal indices of holomorphic sections are positive, we find that ω has no zeroes; inother words, the bundle Λ1,0 is holomorphically trivial.

The space ker(∂L(x)) is the space of meromorphic functions on X which have atworst a simple pole at x. It contains the constant functions, and a meromorphicfunction on X with a single simple pole at x can be viewed as a map f ∶X → CP1.As ∞ is a regular value, f must have degree 1. It follows that for all regular valuesz, f−1(z) must be a single point (this uses the holomorphicity of f). Near a criticalpoint of f of order k, f assumes each value k times, so we conclude that f has nocritical values and therefore is a diffeomorphism, contradicting the assumption thatg = 1. We conclude that ker(∂L(x)) is one-dimensional.

The space ker(∂L(−x)) is the space of holomorphic functions on X which have a

zero at x. As each holomorphic function on X is constant, ker(∂L(−x)) = 0.

Exercise 4.7.5. Find a canonical divisor of CP1. Prove that each divisor is linearlyequivalent to n ⋅ 0, for a unique n ∈ Z. Compute dimH0(X,D) by hands and verifythe Riemann-Roch theorem by hands.

The proof of Riemann-Roch that we gave used the main theorem on ellipticregularity and the theory of characteristic classes as the main ingredients. Whilethe characteristic class theory was overkill (in fact, we only needed the first Chernclass only, and that can be done in an easier way), the use of the regularity theoremis, most emphatically, neccessary. In Riemann surface texts, the analysis goinginto the Riemann-Roch theorem is a version of the general theory (which can besomehow simplified, but is still difficult).

If one knows in advance thatX is a projective variety (i.e. a complex submanifoldof CPn for some n), then it is known (Chow’s theorem) that X is algebraic and inthis case, there is a purely algebraic proof of Riemann-Roch (GAGA). In fact, eachRiemann surface can be embedded into projective space, and this is a consequenceof Riemann-Roch!

4.8. Literature. The Hodge decomposition theorem is a classical result, and eachsource that discusses elliptic operators on closed manifolds gives a proof of theHodge theorem: [31], [68], [70], [48], [27] and others. The proof of the Riemann-Roch theorem is my own. I do not claim that this proof is particularly simple,but it has two advantages over the standard proofs given in Riemann surface texts,such as [25] or [31]. The first one is that the standard proof only works for linebundles; whereas our proof works in a more general context. This leads to thesecond advantage: it motivates the introduction of K-theory as the central playerin index theory. In fact, the proof of the general index theorem on the sphere (animportant special case, proven as Theorem 10.8.8) follows the same pattern.

68 JOHANNES EBERT

5. Some bundle techniques

5.1. Vector bundles. The definition of a vector bundle won’t be repeated here.We work with vector bundles over the field K = R,C. There are two categories: dif-ferentiable vector bundles over smooth manifolds, and topological vector bundlesover Hausdorff spaces. All theorems will hold in both categories. We formulate ev-erything for topological bundles, replacing the words ”topological space” by ”man-ifold”, ”vector bundle” by ”smooth vector bundle” and ”continuous” by ”smooth”gives a valid argument. When π ∶ V → X is a vector bundle, we denote the fibresby Vx ∶= π−1(x).

Definition 5.1.1. Let V → X be a vector bundle. A subbundle W ⊂ V is a unionof sub vector spaces of the fibres W = ∐x∈XWx, such that W is locally trivial inthe subspace topology.

There are several types of bundle maps. Unfortunately, there is no suggestiveterminology. The most general notion is when W → Y , V → X are vector bundlesand f ∶X → Y is a continuous map. One considers maps ϕ ∶ V →W such that

V

ϕ // W

X

f // Y

commutes and such that ϕ is fibrewise linear (and continuous). We call such anϕ a bundle morphism over f . Two special cases are important enough to deserve aname on its own:

(1) If f is the identity map on X, we call ϕ a vector bundle homomorphism.(2) If f is arbitrary, but ϕ ∶ Vx →Wf(x) is an isomorphism for each x ∈X, then

ϕ is called a bundle map over f .

If you know a better name for these things, please let me know. The pullback ofvector bundles has the following universal property. Let f ∶X → Y and π ∶W → Y

be a vector bundle. Then there is a bundle map f ∶ f∗W ∶= (x,w) ∈X ×W ∣f(x) =π(w)→W over f , defined by f(x,w) = w. Assume that V →X is another bundleand φ ∶ V → W be a bundle morphism oder f . Then there is a unique bundle

homomorphism ϕ ∶ V → f∗W such that f ϕ = φ.

Lemma 5.1.2. Any vector bundle over a paracompact base space admits a bundlemetric.

Lemma 5.1.3. Let F ∶ V →W be a vector bundle homomorphism which is bijective.Then F is an isomorphism of vector bundles, i.e. F −1 is continuous.

Proof. This follows from the fact that the inversion map on GLn(K) is differen-tiable.

Lemma 5.1.4. A subbundle W ⊂ V has adapted charts, i.e for each x ∈ X, thereis a neighborhood U and a bundle chart V ∣U ≅ U ×Kn that sends W ∣U to U ×Km.

Proof. The problem is a local one, which is why we can assume that V = X ×Kn.Let o ∈ X, and let Uo ⊂ Kn be a complement of Wo. Consider F ∶ W ⊕ Uo → V ,(w,u) ↦ w + u; a bundle homomorphism. F is an isomorphism at o, and so it


is for all x in a neighborhood U of o. By Lemma 5.1.3, F gives an isomorphismW ∣U ⊕Uo ≅ V ∣U . The inverse is the desired adapted chart.

Corollary 5.1.5. Let V ⊂ X × Kn be a subbundle and let Px be the orthogonalprojection onto Vx. Then X ↦Matn,n(K), x↦ Px is continuous.

Proof. The problem is local, so assume V ≅ X × Km. This isomorphism definessections s1, . . . , sm of V which are everywhere linear independent. Applying theGram-Schmidt process to s1, . . . , sm defines an orthonormal basis t1, . . . , tm of V .The inclusion V → X ×Km is given by a continuous function A ∶ X → Matn,m(K)that takes values in the matrices A such that A∗A = 1m. The orthogonal projectionis P = AA∗.

With the same technique, one can prove.

Lemma 5.1.6. Let F ∶ V → W be a vector bundle homomorphism, and assumethat x↦ rank(Fx) is constant. Then ker(F ) is a vector bundle.

Proof. Again, the problem is local, so we can assume that V and W are trivialand F is given by a continuous function F ∶ X → Matm,n(K). Let o ∈ X and Pthe orthogonal projection onto ker(Fo). Consider G = F ∗F + P . At o, G is anisomorphism, so it is for nearby x ∈ U ⊂ X. By definition, Gx maps ker(Fx) toker(Fo). Since Gx is an isomorphism and the dimensions of ker(Fo) and ker(Fx)agree, Gx ∶ ker(Fx) → ker(Fo) is an isomorphism. It follows that G is a bundleisomorphism over U that maps ker(F ) to X × ker(Fo) and hence it reveals ker(F )as a subbundle.

Corollary 5.1.7. (1) The orthogonal complement of a subbundle is again avector bundle.

(2) The image of a vector bundle homomorphism with constant rank is a vectorbundle.

Proof. Let W ⊂ V be a subbundle. Equip V with a bundle metric. Let P ∶ V → Vbe the orthogonal projection onto W . By looking at adapted charts, one sees that Pis a continuous bundle homomorphism. The orthogonal complement W is ker(P ),which by Lemma 5.1.6 is a subbundle. For the second part, pick bundle metricsand observe that Im(F ) = ker(F ∗), which by the first part and Lemma 5.1.6 is avector bundle.

The most important vector bundle is the tautological bundle.

Definition 5.1.8. The Grassmann manifold of k-dimensional subspaces of KnGrk(Kn) is the set of all k-dimensional subspaces of Kn. We identify Grk(Kn)with the set P ∈ Matn,n(K)∣P 2 = P ;P ∗ = P, rank(P ) = k, by sending a subspaceV ⊂ Kn to the orthogonal projection onto it (and by sending a projection onto itsimage). Let Vk,n ⊂ Grk(Kn)×Kn be the set of all pairs (V, v), V ∈ Grk(Kn), v ∈ V .

The Grassmann manifold is compact (it is a closed bounded subset of the spaceof matrices), and we will see soon that it is indeed a manifold.

Lemma 5.1.9. Vk,n is a subbundle of the trivial vector bundle.

Proof. Vk,n is the image of the canonical homomorphism (P, v) ↦ (P,Pv) of thebundle Grk(Kn) ×Kn.

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Proposition 5.1.10. Let X be a space. Then there is a bijection between the setof k-dimensional subbundles of X ×Kn and continuous maps X → Grk(Kn). Thebijection sends a bundle V ⊂ X × Kn to the map x ↦ Vx; the other direction isf ↦ f∗Vn,k.

If V → X is a vector bundle, then bundle monomorphisms V → X ×Kn are inbijection with pairs (f, a), f ∶X → Grk(Km) and a ∶ V ≅ f∗Vk,n.

Proof. Let V be a subbundle. The map fV ∶ X → Grk(Kn), x ↦ Vx is continuous,by Lemma 5.1.5. Clearly both bijections are mutually inverse.

One should think of a vector bundle as a family of vector spaces that dependscontinuously on a space X. This proposition gives some first credibility that thisway of thinking is indeed accurate. There are two steps missing: first we need toshow that any vector bundle is indeed isomorphic to a subbundle of a trivial vectorbundle, and this isomorphism needs to be canonical in a reasonable way. Second,we want to prove that homotopic maps give rise to isomorphic vector bundles.

Theorem 5.1.11. Let F ∶ [0,1]×X → Y be a homotopy from F0 to F1 and V → Ybe a vector bundle. If X is paracompact, then F ∗

0 V ≅ F ∗1 V .

Before we give the proof, let us collect a technical lemma.

Lemma 5.1.12. Let Vi → X, i = 0,1, be vector bundles over a paracompact Haus-dorff space and let A ⊂ X be closed. Assume that V0∣A ≅ V1∣A. Then there exists aneighborhood U of A and a bundle isomorphism V0∣U ≅ V1∣U .

Proof. Let φ ∶ V0∣A → V1∣A be an isomorphism. We can find open sets Ui ⊂X, i ∈ I,0 /∈ I that cover A, such that Vj ∣Ui is trivial and take U0 = X −A as another openset. Let λi be a partition of unity subordinate to the covering. Since paracompactspaces are normal, the space Ui is normal. The restriction of φ to Ui is given by afunction Ui ∩A→ GLn(K), with respect to some unnamed bundle charts.

By Tietze’s extension theorem, we can find extension φi of φ∣A∩Ui over Ui. φi isonly a bundle homomorphism, not an isomorphism. Put ψ = ∑i∈I λiφi. This is abundle homomorphism, and an isomorphism over A. Since being an isomorphismis a local condition, ψ is an isomorphism over some neighborhood U of A.

Remark 5.1.13. In the differentiable case, there are two types of extension prob-lems one could consider. If A ⊂X is an arbitrary closed subset, one calls a functionA → R smooth if it is the extension of a smooth function on some neighborhoodof A. In this case, the statement of Lemma 5.1.12 is vacuous. The other relevantcase is when A ⊂ X is also a submanifold. In that case, one has to use a tubularneighborhood of A in X.

Proof, under the additional assumption that X is compact. Let jt ∶ X → [0,1] ×Xbe the inclusion x ↦ (t, x). Since Ft = F jt, it is enough to prove the theoremwhen F is the identity, viewed as a homotopy from j0 to j1. In other words, weassume that V → [0,1] × X is a vector bundle, and let Vt ∶= j∗t V . We want toshow that V0 ≅ V1. To this end, we introduce an equivalence relation ∼ on [0,1]:t ∼ s iff Vt ≅ Vs. Of course, this is an equivalence relation. Once we prove that theequivalence classes are open, we are done, since [0,1] is connected. Fix t ∈ [0,1]and consider the bundles Vt × [0,1] and V over X × [0,1]. By definition, theirrestrictions to X × t are isomorphic. By Lemma 5.1.12, we find a neighborhood


X×t ⊂ U ⊂X×[0,1] over which these two bundles are isomorphic. By compactness,U contains a strip X × (t − a, t + b). Hence if s ∈ (t − a, t + b), then Vt ≅ Vs.

The general case needs similar ideas, but with more care. I recommend to readthe proof in [64] in greater generality.

Theorem 5.1.14. Let X be a compact space and π ∶ V → X be a vector bundle ofrank k. Then there exists n >> 0 and an injective bundle homomorphism φ ∶ V →X × Kn. If moreover A ⊂ X is closed and ψ ∶ V ∣A → A × Km is an already givenbundle monomorphism, then we can pick φ to coincide on A with in,m ψ, wherein,m ∶ X × Km → X × Kn (the price one has to pay is that m is potentially verylarge).

Proof. Let Ui, i = 1, . . . , r, be an open cover, (π;hi) ∶ V ∣Ui ≅ Ui ×Kk bundle trivi-alizations and λi be a partition of unity subordinate to this cover. Let n = rk anddefine φ ∶∶ V →X × (Kk)r by

φ(v) ∶= (π(x), λ1(π(v))φ1(v), . . . , λr(π(v))φr(v)).This is a bundle injection, as one checks easily. For the relative case, let U0

be a neighborhood of A and (π,φ0) ∶ V ∣U0 → X × Kn be an extension of ψ toa bundle homomorphism, as guaranteed by Lemma 5.1.12. Since being injectiveis an open condition, we can assume that (π,φ0) is injective (after making U0

smaller). Let (π,φ1) ∶ V → X × Km be an embedding as just constructed. Letµ be a function which is equal to 1 on A and has support in U0. Let φ(v) ∶=(π(v), µ(π(v))φ0(v), (1 − µ(π(v)))φ1(v)), which is the desired extension.

Corollary 5.1.15. For each vector bundle V → X over a compact space, there isa bundle V →X such that V ⊕ V ≅X ×Cn.

Remark 5.1.16. For compact manifolds, we can use the same argument. It is alittle surprising that in the smooth case, the compactness of X is not necessary.More precisely, if V →M is a smooth vector bundle, we can embed V into M ×Rm,for some large m. The reason is the Whitney embedding theorem. Since V is amongother things a manifold, we can find an embedding of manifolds j ∶ V → Rm. Thedifferential dj ∶ TV → V × Rm is an everywhere injective homomorphism of vectorbundles over V . But the restriction of TV to the zero section is nothing else thanTM ⊕ V , so we can produce the desired embedding.

If V is complex, we first take a real embedding f ∶ V →X ×Rm ⊂ Cm and define

a C-linear embedding by f(v) = f(v) − if(iv).

We can now formulate and prove the classification theorem for vector bundles.Let X be a compact space and [X,Grk(Kn)] be the set of homotopy classes. More-

over, VectkK(X) is the set of isomorphism classes of rank k vector bundles over X.If f ∶X → Grk(Kn), we can form f∗Vk,n →X. The isomorphism class of this vectorbundles does not depend on f , by Theorem 5.1.11. So we get a well-defined map

[X,Grk(Kn)]→ VectkK(X).There is no n on the right hand side. In fact, there is an inclusion i ∶ Grk(Kn)→

Grk(Kn+1), and i∗Vk,n+1 ≅ Vk,n. Thus, by making n larger and larger, we obtain amap

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colimn[X,Grk(Kn)]→ VectkK(X).

Theorem 5.1.17. For each compact Hausdorff space X, the above map is a bijec-tion.

Proof. Since any vector bundle can be embedded into X×Kn, the map is surjective.Let fi ∶ X → Grk(Kni) be two maps, which represent two elements in the colimitthat go to the same vector bundle V . We can take both fi to go to Grk(Kn),n ≥ n0, n1. What this means is that there are isomorphisms ai ∶ V ≅ f∗i Vk,n. Inother words, we have two bundle maps ji ∶ V → X × Kn, which is the same as abundle map of V × 0,1→X × [0,1]×Kn. By Theorem 5.1.14, we can extend thisto a bundle map of j, after increasing n. This means that f0 and f1 are homotopic,after increasing n.

If X is compact, then colimn[X,Grk(Kn)] ≅ [X, colimnGrk(Kn)] (a property ofthe colimit topology). Thus we have shown that there is a bijection

[X,Grk(K∞)]→ Vectk(X).

5.2. Principal bundles. A more flexible jargon to talk about bundles is providedby the theory of principal bundles. Let us briefly recall the notion of fibre bundle.

Definition 5.2.1. A fibre bundle over a space X is a map π ∶ E → X so that foreach x ∈ X, there exists a neighborhood U of x and a homeomorphism π−1(U) ≅U × π−1(x) over U . The space π−1(x) =∶ Ex is called the fibre over x.

At least if X is connected, then all fibres are homeomorphic. Sometimes, we saythat π ∶ E → X is a fibre bundle with fibre F , if all fibres are homeomorphic toF . But there is a danger in this notion, because it invites the reader to identifyall fibres with each other, which will inevitably get you into hot water. This isbecause there are several ways of identifying the fibres Ex with F . The theory ofprincipal bundles provides a precise calculus to keep track of all identifications andif you understand it, you have taken a big psychological hurdle when dealing withbundles.

Definition 5.2.2. Let G be a topological group and X a space. A G-principalbundle consists of a right G-space E and a continuous map π ∶ E → X, such thatthe following condition holds: For each x ∈ X, there is a neighborhood U and ahomeomorphism φ ∶ π−1(U) → U × G, such that prG φ = π. Moreover, φ is Gequivariant when U ×G is equipped with the action (x,h)g ∶= (x,hg).

A bundle map E → E′ of G-principal bundles (possibly over different spaces) isa G-equivariant map. Any bundle map covers a map f ∶ X → X ′. If the bundlemap covers the identity, it is bijective and we say that the bundle map is a bundleisomorphism.

It can be shown that a bijective bundle map is a homeomorphism, and thisjustifies our usage of the word ”isomorphism”.j In our applications, G will be a Liegroup. Requiring that E and X are smooth manifolds, the action of G and π andφ being smooth, one arrives at the notion of a smooth principal bundle. There isa notion of pullback: if f ∶ Y → X is a map and π ∶ E → X a G-principal bundle,then f∗E ∶= (y, e) ∈ Y ×E∣f(y) = π(e) has the natural structure of a G-principal


bundle. If you are a novice in bundle theory, you are invited to provide the detailsas an exercise.

Exercise 5.2.3. Let X be a connected space that has a universal covering X →X.Equip X with the structure of a π1(X,x)-principal bundle (this is irrelevant forindex theory).

The most substantial example of a principal bundle is the frame bundle of avector bundle.

Example 5.2.4. Let V → X be an n-dimensional K-vector bundle. Let Fr(V ) ∶=∐x Iso(Kn, Vx), Iso(Kn;Vx) is the set of all vector space isomorphisms. There isan obvious map π ∶ Fr(V ) → X. The GLn(K)-action is by precomposition: iff ∶ Kn → Vx is an isomorphism and g ∈ GLn(K), then f ⋅ g ∶= f g. If U ⊂ X isopen and φ ∶ U ×Kn → V ∣U be a trivialization, we get a bijection U × GLn(K) →π−1(U) = Fr(V ∣U), namely

(x, g)↦ φx g.The topology on Fr(V ) is the finest one so that all these maps are continuous,

and they are all homeomorphisms.

Exercise 5.2.5. Provide the details of the proof that the above construction givesindeed a GLn(K)-principal bundle. If V is a smooth vector bundle, equip Fr(V )with the structure of a smooth principal bundle.

Remark 5.2.6. A point in Fr(V ) is by definition an isomorphism f ∶ Kn → Vxfor some x. If ei ∈ Kn denotes the ith basis vector, we get a basis (v1, . . . , vn) ofVx, vi ∶= f(ei). Now let g = (gij) ∈ GLn(K). Note that gei = ∑nk=1 gkiek (sic!).Therefore f g(ei) = ∑nk=1 gkivi. So if we view frames as bases of the fibres, theGLn(K)-action becomes (v1, . . . , vn) ⋅ g ∶= (∑nk=1 ak1vk, . . .∑nk=1 aknvk). This mightbe confusing, but it is not bundle theory that is to be blamed, but linear algebra.

Exercise 5.2.7. Prove that local trivializations of a vector bundle V → X are inbijective correspondance with local cross-sections of Fr(V ). More generally, localtrivializations of a principal bundle are in bijection with cross-sections. A principalbundle has a global cross-section iff it is trivial.

Exercise 5.2.8. Let V → M be a rank n vector bundle with Riemannian bundlemetric. Define the O(n)-principal bundle FrO(V ) of orthonormal frames. Sim-ilarly, let V → M be oriented. Define the GLn(R)+-principal bundle of orientedframes.

A principal bundle is, among other things, a G-space E, and the base space Xis the quotient E/G. The G-action is free. The next result is a basic fact in thetheory of Lie groups. The proof can be found in [61]. If G is linear, then you canfind an easier proof in [19], but the details are still quite subtle.

Theorem 5.2.9. Let G be a Lie group and H ⊂ G be a closed subgroup. Then H isa Lie group, the quotient space G/H has the unique structure of a smooth manifoldsuch that G → G/H is smooth, and the quotient map G → G/H is a H-principalbundle.

The power of this result can be explained by some examples. First a lemma.

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Lemma 5.2.10. Let M be a smooth manifold and let G be a Lie group that actstransitively from the left on M . Let x ∈ M and H ⊂ G be the isotropy group of x.Then G/H →M , gH → gx is a diffeomorphism; G →M , g ↦ gx is a H-principalbundle.

Proof. The map p ∶ G → M , g ↦ gx is smooth and H-invariant, p(gh) = p(g).Therefore it descends to a smooth map f ∶ G/H →M which is moreover bijective.We claim that this is a diffeomorphism. Since f is G-equivariant (G acting from theleft!) and the action on both G/H and M is transitive, the rank of df is constant.By Sard’s theorem, f has a regular value, and so f must be a submersion. Sincef is injective, the dimensions have to agree, and so f is a bijective map which haseverywhere full rank, i.e. a diffeomorphism.

Examples 5.2.11.

(1) O(n+ 1) acts on Sn, by rotations. The isotropy group of the vector en+1 isO(n). This shows that O(n + 1)/O(n) ≅ Sn is a diffeomorphism. In fact,we can identify O(n+1) with the total space of the orthogonal frame bundleof TSn (with the usual metric of the sphere).

(2) In a similar way, U(n)→ S2n−1 is a U(n − 1)-principal bundle.(3) CPn is U(n+1)/U(n)×U(1). More generally, the Grassmannian is Grk(Cn) ≅

U(n)/U(k) ×U(n − k).(4) The quotient map S2n+1 → CPn is the unitary frame bundle of the tauto-

logical line bundle.

Corollary 5.2.12. Let G be a Lie group and let G (as a group) act transitively ona set S. Then S has a unique topology and smooth structure so that the G-actionis smooth.

Besides pullbacks, there are a couple of useful constructions with principal bun-dles.

Definition 5.2.13. Let E → X be a G-principal bundle and F → Y be an H-principal bundle. Then E × F → X × Y is a G × H-principal bundle with theproduct action.

The next one is what we call ”change of fibre”, which is very important. If youwant to be comfortable with bundles, you have to absorb this construction.

Definition 5.2.14. Let π ∶ P → X be a G-principal bundle and F a left G-space.The group G acts on the space P × F diagonally, (p, f) ⋅ g ∶= (pg, g−1f). We defineP ×G F ∶= (P × F )/G. The projection map [(p, f)] ↦ π(p) is a well-defined mapP ×G F . Then P ×G F is a fibre bundle with typical fibre F .

The definition comes with a companion.

Definition 5.2.15. A fibre bundle with structural group G and fibre F on Xconsists of a fibre bundle E →X, a G-principal bundle P →X and an isomorphismof fibre bundles P ×G F ≅ E.

It is time to convince ourselves with the use of this construction; there are manyuseful examples.

Example 5.2.16. If V is a vector bundle, then V ≅ Fr(V ) ×GLn(K) Kn. Hence wecould define the notion of a vector bundle by saying that it is a fibre bundle with


structural group GLn(R) and fibre Rn. More generally, if a representation of a Liegroup is given (this is a smooth homomorphism G→ GL(V ) for some vector space)and if P is a G-principal bundle, then P ×GV is a vector bundle. One can formulatethe notion of oriented vector bundle, vector bundle with metric etc. etc. using thiscalculus.

Example 5.2.17. If V → X and W → Y are two vector bundles of rank n, m,then we can form the GLn(K) ×GLm(K)-principal bundle Fr(V ) × Fr(W ) → X ×Y . The Lie group GLn ×GLm acts on Kn ⊕ Km. The resulting bundle Fr(V ) ×Fr(W ) ×GLn(K)×GLm(K) Km+n → X × Y is called the external direct sum. If X = Y ,we can pull back the external direct sum to X with the diagonal X → X ×X andobtain the direct sum.

Exercise 5.2.18. Along the lines of this example, define V ⊗W for two vectorbundles, Hom(V,W ), the dual bundle V ∗, the bundle of alternating and symmetricmultilinear forms and so on.

Example 5.2.19. Consider the principal bundle S2n+1 → CPn. Let S1 ⊂ C act onC by multiplication. The vector bundle S2n+1 ×S1 C→ CPn is the tautological linebundle on CPn.

These examples suggest that when the group action of G on the fibre F preservessome kind of structure on F , we find that the bundle P ×G F has this structure,but now in families. Caution: even though the fibres of a principal bundle looklike groups, they are not. They are right-G-spaces, and do, most emphatically, nothave a multiplication.

Exercise 5.2.20. Formulate the notion of a bundle of: topological groups, finite-dimensional R-algebras, finite-dimensional Lie algebras.

Example 5.2.21. Let V be a complex vector bundle of rank n. The group GLn(C)acts on the complex projective space, in exactly one meaningful way. The bundlePV ∶= Fr(V )×GLn(C)CPn is called the projective bundle to V . Moreover, the groupaction of GLn(C) lifts to an action on the universal line bundle H → CPn and hencegives rise to a line bundle on PV . Prove that the pullback of V to PV splits off aone-dimensional line bundle.

Example 5.2.22. If G → H is a group homomorphism and P → X a G-principalbundle, then P ×GH is in a natural way an H-principal bundle.

Definition 5.2.23. Let G → H be a group homomorphism and Q → X be anH-principal bundle. A reduction of the structural group from H to G consists of aG-principal bundle P →X and an isomorphism of H-principal bundles P ×GH ≅ Q.

Most additional structures that exist on bundles can be expressed using thisnotion. We discuss one example in great detail.

Example 5.2.24. Let V → X be a real n-dimensional vector bundle and let P =Fr(V )→X be the frame bundle. We want to explain that a bundle metric on V is”the same” as a reduction of the structural group of P from GLn(R) to O(n).

Let Q → X be an O(n)-principal bundle and η ∶ Q ×O(n) Rn ≅ V be an isomor-phism. Then η induces an isomorphism

(5.2.25) Q ×O(n) GLn(R) ≅ P.

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Namely, by considering the n columns of a matrix, we get an embedding GLn(R) ⊂(Rn)n. This embedding is O(n)-equivariant, with O(n) acting on GLn(R) by left-multiplication and on (Rn)n by acting on each factor separately. Thus, a point in(Q×O(n)GLn(R))x (x ∈X) gives rise to n vectors in the vector space (Q×O(n)Rn)x,and these vectors are of course linearly independent. Vice versa, from an isomor-phism as in 5.2.25, we obtain an isomorphism

Q ×O(n) Rn ≅ Q ×O(n) GLn(R) ×GLn(R) Rn ≅ P ×GLn(R) Rn ≅ V.Thus a reduction of the structure group of P to O(n) is ”the same” as an O(n)-

principal bundle Q and a vector bundle isomorphism Q ×O(n) Rn ≅ V .If V has a bundle metric, then we let Q be the bundle of orthogonal frames of

V . We can describe Qx either as the set of all orthonormal frames of Vx or as theset of all isometries Rn → Vx. This is an O(n)-principal bundle. There is a bundleisomorphism

Q ×O(n) Rn → V

sending an equivalence class [f, v] ((f, v) ∈ Px × Rn) to f(v) ∈ V . Therefore, abundle metric gives rise to a reduction of the structural group. On the other hand,the bundle Q×O(n)Rn carries a bundle metric. Let [f, v], [f,w] ∈ Q×O(n)Rn. Then,as O(n) preserves the standard scalar product on Rn, ⟨[f, v], [f,w]⟩ ∶= (v,w) doesnot depend on the choice of representative.

We can describe the whole correspondance even more abstractly. Let Q→X bean O(n)-principal bundle and P = Q×O(n)GLn(R). LetM be the set of all positivesymmetric bilinear forms on Rn; this is an open subset of a finite-dimensional vectorspace and hence a manifold which has a GLn(R)-action. In fact, the GLn(R)-actionis transitive. The group O(n) is (by definition) the isotropy group of the elementA0 ∈M (the standard inner product). Therefore M = GLn(R)/O(n) as GLn(R)-space. Moreover, since A0 is O(n)-invariant, it defines a fibre-preserving map

X = Q ×O(n) ∗→ Q ×O(n)M.

The bundle Q ×O(n)M → X is the bundle whose fibre over x is the space of allinner products on the fibre of Q ×R Rn over X. Therefore, a section of M → X isa bundle metric.

Example 5.2.26. Let V be a vector bundle and W ⊂ V be a subbundle, of ranksn < m. Let Gm,n be the group of all linear transformations of Km that map Knto itself. Show that the frame bundle Fr(V ) admits the reduction of the structuregroup to Gm,n and show how to construct the subbundle W , the bundle V and thequotient bundle V /W out of this reduction.

Example 5.2.27. A reduction of the structural group from GLn(R) to GLn(R)+is the same as an orientation, to O(n + 1) is the same as a bundle metric, and soon.

Finally, we briefly indicate how the classification theory of principal bundlesworks.

Theorem 5.2.28. Let F ∶X×[0,1]→ Y be a homotopy and P → Y be a G-principalbundle. If X is paracompact, then F ∗

0 P ≅ F ∗1 P .


Theorem 5.2.29. Let G be a topological group. Then there exists a ”universalG-principal bundle” EG → BG, such that for each paracompact space X, there isa bijection [X,BG] ≅ PrinG(X). The bundle EG → BG is unique up to homotopyequivalence, and it is characterized by the property that EG is contractible.

Examples 5.2.30. Let G = GLn(K). The frame bundle of the tautological vectorbundle over Grn(Kk) is the Stiefel manifold Stn,k(K). The colimit colimk Stn,k(K)is contractible and this shows that EGLn(K) = colimk Stn,k(K), in accordance toour previous classification theory.

If G ⊂ GLn(R) is a closed subgroup, we could take EG ∶= EGLn(R) and BG ∶=EGLn(R)(G) = EGLn(R) ×GLn(R) GLn(R)/G.

5.3. Literature. The language of bundles is commonly used in topology and dif-ferential geometry, and any serious student of these areas will have to speak itfluently sooner or later. I cannot give a single reference; the classic source is [62].The book [40] is still popular. A more concise place to learn this material is [64].

78 JOHANNES EBERT

6. More on de Rham cohomology

The remaining goal for this term is the proof of the Gauß-Bonnet-Chern theoremand the Riemann-Roch theorem, which are the role models for the general indextheorem.

In both cases, the index theorem will take the following form. The Gauß-Bonnet-Chern theorem states that there is, for an oriented manifold, a specific cohomologyclass e(TM) ∈ Hn(M) such that ind(D) = χ(M) = ∫M e(TM). The class e(TM)is the Euler class. Riemann-Roch will be a similar formula.

For a while, we will abandon the differential operators, Sobolev spaces and esti-mates and focus on the right-hand side of the index theorem. We develop the theoryof characteristic classes. We do this in the framwork of de Rham cohomology, andwe will actually do it twice. There is the global theory of characteristic classes,which is an offspring of Poincare duality. And there is the local theory (curvature).

We reveal a close connection between differential forms and the global geometryof a manifold. The tools we develop will be crucial to the proof of the Gauß-Bonnet-Chern theorem and also for the later translation of the index formula fromK-theory to cohomological terms. There is also an inherent beauty! But as oftenin mathematics, beauty and elegance needs support by strong workhorses.

In singular (co)homology theory, there are two main technical workhorses: rel-ative cohomology and the pairing between cohomology and homology. We have toreplace these pillars by something.

6.1. Technical prelimiaries. The fundamental property of the de Rham coho-mology is its homotopy invariance. We recall how the proof works because we willneed to see that it can be modified to prove homotopy invariance of some versionsof de Rham cohomology. Let M be a manifold. One defines an operator

P ∶ Ap(M × [0,1])→ Ap−1(M)by setting

Pω ∶= ∫1

0j∗t (ι∂tω)dt.

Explanations: ∂t is the vector field pointing in the [0,1]-direction; jt ∶ M →M ×[0,1] is jt(x) = (x, t). The form j∗t (ι∂tω) is a p−1-form on M , and t↦ j∗t (ι∂tω)is a smooth curve in the vector space Ap−1(M), and the integral is the usualLebesgue integral for functions with values in topological vector spaces. In localcoordinates, one shows that Pd+dP = j∗1 −j∗0 , and therefore P is a chain homotopy.

We need to fix an orientation convention, once and for all.

Convention 6.1.1. If M and N are oriented manifolds, we orient the productby the following requirement. If (v1, . . . , vm) is an oriented basis for TxM , and(w1, . . . ,wm) an oriented basis for TyN , the (v1, . . . , vn,w1, . . . ,wm) is an orientedbasis of T(x,y)M ×N .

If V →M is an oriented vector bundle over an oriented manifold, then we orientthe total space by saying that an oriented chart V ∣U ≅ U × Rn is an orientationpreserving diffeomorphism of manifolds.

If W,V → X are two oriented vector bundles, we orient V ⊕W by saying thatan oriented basis of Vx, followed by an oriented basis of Wx, is an oriented basis ofVx ⊕Wx.


If N ⊂M is a submanifold, then there is an almost natural isomorphism TM ∣N ≅TN ⊕ νMN , where νMN = TM ∣N /TN denotes the normal bundle of N in M . So ifM is oriented, then, according to the convention for sums of vector bundles, anorientation of N determines an orientation of the normal bundle and vice versa.

Definition 6.1.2. Let M be a manifold. By the symbol A∗c(M), we denote thespace of compactly supported differential forms. It is clear that this is a chaincomplex and an ideal in A∗(M). H∗

c (M) is the cohomology of this chain complex,the compactly supported cohomology of M .

Using this new cohomology, we obtain a new level of flexibility when dealingwith cohomology, but there are some pitfalls. If f ∶M → N is a smooth map, thenwe do not have in general a map f∗ ∶ A∗c(N) → A∗c(M) (let f ∶ R → ∗ to see whatgoes wrong). But if f is a proper map, we have a pullback f∗.

There is another functoriality: if U ⊂ M is an open subset, we get a mapA∗c(U) → A∗c(M), and of course there is a map A∗c(M) → A∗(M), which some-times carries important information as well. Recall that in singular (co)homology,relative cohomology is a central technical tool. Here is our replacement for it:

Definition 6.1.3. Let M be a manifold and let A ⊂ M be a closed subset. Wedefine A∗(M)A ∶= colimA⊂U A∗(U), the space of germs of forms near A.

The technical environment that makes cohomology theory breathe is homolog-ical algebra, and in particular exact sequences. There are three important exactsequences in de Rham theory. First, there are two Mayer-Vietoris-sequences. LetU,V be open. Then there are sequences

0→ A∗(U ∪ V )→ A∗(U)⊕A∗(V )→ A∗(U ∩ V )→ 0

ω ↦ (ω∣U , ω∣V ); (ω, η)↦ ω∣U∩V − η∣U∩Vand

0→ A∗cpt(U ∩ V )→ A∗cpt(U)⊕A∗cpt(V )→ A∗c(U ∪ V )→ 0

ω ↦ (ω,−ω); (ω, η)↦ ω + η.That the second is exact is obvious (!). The first one is slightly more complicated

and involves partitions of unity, see [19], p. 287 f. In cohomology, we obtain twoMayer-Vietoris sequences.

Lemma 6.1.4. Assume that M ∖A is relatively compact in M . Then there is anexact sequence

0→ A∗c(M ∖A)→ A∗(M)→ A∗(M)A → 0.

If A is either a submanifold or a codimension 0 submanifold (with boundary),then the restriction A∗(M)A → A∗(A) is a quasiisomorphism.

Proof. It is clear that the first map is injective and that the composition is zero.If ω ∈ A∗(M) maps to zero, it means that there is a neighborhood A ⊂ U such

that ω∣U = 0, whence ω has support in M −U and this is compact. An element inA∗(M)A is represented by a form ω on U ⊃ A. Pick a function µ that is 1 near A

80 JOHANNES EBERT

and has support in U , then we can extend µω to a form on M , which representsthe same form in the colimit.

For the second part, first note that for an open neighborhood U of A, A∗(U)A =A∗(M)A. We use the tubular neighborhood theorem [20]. Let U be a tubular neigh-borhood and r ∶ U → A be the projection. The map A∗(M)A → A∗(A) is surjectivesince for any η, the form r∗η represents a preimage. Also, since r is a homotopyequivalence, the composition A∗(U) → A(U)A → A∗(A) is a quasiisomorphism,and therefore A(U)A → A∗(A) is surjective in cohomology. Thus it remains toprove that the first map A∗(U) → A∗(U)A is surjective in cohomology, and it isenough to show that for each cohomology class of A∗(U)A, there is a smaller tubu-lar neighborhood W such that the class comes from A∗(W ), because all tubularneighborhoods are homotopy equivalent. So let ω ∈ H∗A∗(M)A. It is representedby a closed form on a certain V ⊃ A. Take a tubular neighborhood U ⊂ V and acut-off function η which is 1 on U and has support in V . The form ηω is not closedon V , but its restriction to U is closed.

Lemma 6.1.5. For k ≠ n, Hkc (Rn) = 0. The integration homomorphism Hn

c (Rn)→R, ω ↦ ∫Rn ω is an isomorphism.

Proof. The case n = 0 is trivial, and we assume n > 0. We consider the 1-pointcompactification Sn of Rn. From Lemma 6.1.4, we get the exact sequence 0 →A∗c(Rn)→ A∗(Sn)→ A∗(Sn)∞ → 0 and the fact that A∗(Sn)∞ is quasiisomorphicto A∗(∗). The long exact cohomology sequence has the following outlook

H∗c (R) // H∗(Sn)

epiyyH∗(∗)

[+1]

dd

The symbol [+1] reminds you of the degree shift, and the map H∗(Sn)→H∗(∗)is surjective, since it is induced from the inclusion ∗ = ∞ → Sn, and splits via theconstant map Sn → ∗. Thus the cohomology sequence falls apart as

0→Hkc (Rn)→Hk(Sn)→Hk(∗)→ 0.

Together with the known computation of H∗(∗) and H∗(Sn), this proves thatHn(Rn) is one-dimensional (and Hk

c (Rn) = 0 for k ≠ n). For the statement aboutthe integration homomorphism, it is enough to find a single compactly supportedclosed n-form on Rn with integral 1, which is easy: take a suitable bump functiona and let ω = a(x)dx1 ∧ . . . ∧ dxn.

We need another preliminary, namely a flexible local-to-global principle.

Proposition 6.1.6. (The bootstrap lemma) Let M be a manifold, and let P (U) bea statement about open subsets of M . Suppose:

P (∅) is true, There is a cover (Ui)i∈I of M , so that if V that is contained in one of the Ui

and is diffeomorphic to an open convex subset of Rn, then P (V ) is true3. If P (U), P (V ) and P (V ∩U) are true, then so is P (U ∪ V ).

3This precise formulation is quite useful in later applications


If Un are disjoint open subsets and if P (Ui) is true for all i, then P (∪iUi)is true.

The proof can be found in [19], Lemma V.9.5, with a slightly weaker assumption.It is easy to generalize the proof.

6.2. Poincare duality - again! -, Kunneth theorem and the Thom isomor-phism. The Poincare duality theorem was the main result of the class ”TopologyII” and if you attended the class, you remember that among all the results in thisclass, this was by far the most difficult one. We gave a proof for closed manifoldsusing elliptic regularity theory. In de Rham theory, there is a slick and short proof.What should the theorem - for a noncompact manifold - look like? Let Mn be anoriented manifold. If ω ∈ Ap(M) and η ∈ An−pc (M), we form

I(ω, η) ∶= ∫Mω ∧ η ∈ R.

The integral is well-defined because ω has compact support. It is easy to seethat I(ω, η) depends, when the forms are closed, only on the cohomology classes inHpc (M) and Hn−p(M). Thus we get a bilinear map

I ∶Hp(M) ×Hn−pc (M)→ R

and therefore two maps

D ∶Hp(M)→Hn−pc (M)∨; E ∶Hn−p

c (M)→Hp(M)∨.

We used the symbol ∨ for the dual space, in order to avoid having to manysymbols ∗ floating around.

Theorem 6.2.1. (Poincare duality - de Rham version) For any oriented n-manifoldM , the map D ∶Hn−p(M)→Hp

c (M)∨ is an isomorphism.

We explicitly do not assert that the other map E ∶Hpc (M)→Hn−p(M)∨ is an iso-

morphism. There is an asymmetry between cohomology and compactly supportedcohomology, and this has a real reason, rooted in - set theory.

Here is a counterexample. Let M be a countably infinite discrete set (a manifoldof dimension 0). It is easy to see that H0

c (M) ≅ R∞ (the dimension is ℵ0), whileH0(M) has dimension 2ℵ0 . Under a suitable finiteness condition, the other map isan isomorphism as well, see below.

Proof. The first step consists of sign conventions. Let A∗c(M)∨ be the dual chaincomplex. The differential is

δ ∶ An−pc (M)∨ → An−p−1c (M)∨; δ(`)(η) ∶= (−1)p+1`(dη)

and the sign is chosen so that D ∶ A∗(M)→ An−∗(M)∨ is a chain map.We use the bootstrap lemma. If M = ∅, there is not much to show. For M = Rn,

we know H∗(Rn) and H∗c (Rn) by Lemma 6.1.5. Clearly, the constant form 1

goes under D to the integration homomorphism, which is a nonzero element in a1-dimensional vector space.

If U ⊂ V is an open subset, then the diagram

82 JOHANNES EBERT

Ap(V )

D

// Ap(U)

D

An−pc (V )∨ // An−pc (U)∨

(the horizontal maps are induced by restriction) is commutative, which is trivialto verify. Therefore, for two open sets U0, U1 ⊂M , the following diagram of chaincomplexes and chain maps (with exact rows) commutes:

0 // A∗(U0 ∪U1)

DU0∪U1

// A∗(U0)⊕A∗(U1)

DU0⊕DU1

// A∗(U0 ∩U1)

DU0∩U1

// 0

0 // An−∗(U0 ∪U1)∨ // An−∗(U0)∨ ⊕An−∗(U1)∨ // An−∗(U0 ∩U1)∨ // 0.

If DU0 , DU1 and DU0∩U1 are quasiisomorphisms, then, using the long exact co-homology sequence of the above sequences and the 5-lemma, it follows that DU0∪U1

is a quasiisomorphism.The last hypothesis of the bootstrap lemma is to verify that if Ui, i ∈ I, are

disjoint open subsets of M and DUi is a quasiisomorphism for all i ∈ I, then DU is aquasiisomorphism where we put U =∐i∈I Ui. This is because H∗(U) =∏i∈I H

∗(Ui)and H∗

c (U) =⊕i∈I H∗c (Ui) and because taking dual spaces converts direct sums into

direct products (this is also the reason for the asymmetry).

Proposition 6.2.2. The following conditions on a manifold M are equivalent:

(1) H∗(M) is finite dimensional.(2) H∗

c (M) is finite dimensional.(3) The other duality homomorphism E is an isomorphism as well.

Here we take H∗ = ⊕p≥0Hp, which is a finite sum. A manifold with these

properties is called of finite type. Each closed manifold is of finite type.

We need a fact from linear algebra:

Lemma 6.2.3. Let V be a real vector space. Then V is finite dimensional if andonly if the natural map ι ∶ V → V ∨∨ to the dual space is an isomorphism.

This is proven in [18], chapter II, §7.5 Theorem 6.

Proof of Proposition 6.2.2. The easy parts of the Proposition are:

If M is compact, then H∗(M) = H∗c (M), and therefore E agrees with D.

So for compact M , E is an isomorphism. If H∗(M) is finite-dimensional, then so is the dual space of H∗

c (M), andhence H∗

c (M) itself. If H∗

c (M) is finite-dimensional, then so is H∗(M) ≅H∗c (M)∨.

It remains to be shown that E is an isomorphism iff H∗(M) is finite dimensional.To see this, consider the diagram


H∗(M) ι //

D

&&

H∗(M)∨∨

E∨

Hn−∗c (M)∨,

which is commutative, as one checks easily. If H∗(M) is finite-dimensional, thenι is an isomorphism and hence so is E∨. But then E has to be an isomorphism(taking duals is an exact functor). Vice versa, if E is an isomorphism, then so is ι,which means, by the lemma, that H∗(M) is finite-dimensional.

Corollary 6.2.4. If M is compact, then Hk(M) is finite dimensional.

Now we pass to the Kunneth theorem, which expresses the cohomology of aproduct in terms of the cohomologies of the product. The Kunneth theorem onlyholds under a finiteness assumption or for compactly supported cohomology. LetM , N be two manifolds and let prM ∶ M ×N → M and prN ∶ M ×N → N be theprojections. We define the exterior product of two forms ω on M and η on N by

ω × η ∶= pr∗Mω ∧ pr∗Nη

(one can recover ω ∧ η ∶= ∆∗(ω × η), using the diagonal ∆M ∶ M → M ×M).One the tensor product A∗(M)⊗A∗(N), we introduce the differential ∂(ω ⊗ η) ∶=(dω)⊗ η + (−1)∣ω∣ω ⊗ dη, so that

A∗(M)⊗A∗(N)→ A∗(M ×N) and A∗c(M)⊗A∗c(N)→ A∗c(M ×N)

are chain maps. The second one induces a map

(6.2.5) ⊕p+q=k

Hpc (M)⊗Hq

c (N)→Hkc (M ×N).

Theorem 6.2.6. (Kunneth theorem) The map 6.2.5 is an isomorphism, for allmanifolds M and N .

If M and N are of finite type, one can derive that

⊕p+q=k

Hp(M)⊗Hq(N)→Hk(M ×N)

is an isomorphism (we do not try to state the most general assumption here).

Proof of the Kunneth theorem. We use the bootstrap lemma, but proceed in twosteps. First assume that N = Rn and let M vary. For the case M = Rm, useLemma 6.1.5 and Fubini’s theorem. The other hypotheses of the bootstrap lemmaare verified by the same ideas as in the proof of the Poincare duality theorem. Forthe Mayer-Vietoris property, use that taking tensor products is an exact functor(because we work over a field). For the countable disjoint union property, use thattensor products commute with direct sums.

The second step is the general case and uses the same argument.

84 JOHANNES EBERT

The next fundamental result we need is the Thom isomorphism theorem. Letπ ∶ V → M be a vector bundle. Let Apcv(V ) be the space of forms with verticallycompact support (i.e the map π ∶ supp(ω) →M is proper). This is clearly a chaincomplex and we have a Mayer-Vietoris sequence, in the sense that if U0, U1 ⊂ Mare open and Vi = V ∣Ui , then there is an exact sequence of chain complexes

0→ A∗cv(V0 ∪ V1)→ A∗cv(V0)⊕A∗cv(V1)→ A∗cv(V0 ∩ V1)→ 0.

Moreover, the complex of vertically compactly supported forms is contravariantfor bundle maps V → W , and homotopic bundle maps induce chain homotopicmaps.

Exercise 6.2.7. Prove these assertions. Hint: go to the proof of homotopy invari-ance of de Rham cohomology and modify the details.

Definition 6.2.8. Let V → M be an oriented vector bundle of rank n. A Thomform is a closed form τ ∈ Ancv(V ) such that

(6.2.9) ∫Vxτ = 1.

holds for each x ∈M . A Thom class is the cohomology class of a Thom form.

Theorem 6.2.10. Each oriented vector bundle π ∶ V →M has a Thom form.

Proof. First, we assume that the base space M is compact, k-dimensional andoriented. Then V is oriented as a manifold. Moreover, V is homotopy equivalent toM , and thus it has finite type. Moreover, A∗cv(V ) = A∗c(V ). By Poincare duality,there are isomorphisms

Hncv(V ) =Hn

c (V )E≅ Hk(V )∨ ≅Hk(M)∨.

A closed form α ∈ A∗cv(V ) is mapped, under these isomorphisms, to the linearform

η ↦ ∫Vπ∗η ∧ α.

On the other hand, we have a distinguished element in Hk(M)∨, the integrationhomomorphism J ∶ Hk(M) → R. We pick τ so that it maps to J . In other words,for each closed k-form η on M , we have

(6.2.11) ∫Vπ∗η ∧ τ = ∫

Mη.

We claim that τ is a Thom form, in other words, the integral ∫Vx τ has value 1 for

each x ∈M . We pick an oriented coordinate chart x ∶ U → Rk on M and an arbitraryk form η on U with integral 1 and compact support in U . Moreover, we pick orientedbundle coordinates ξ on V . The form η can be written as a(x)dx1 ∧ . . . ∧ dxk, andτ can be written as

b(x, ξ)dξ1 ∧ . . . dξn + ζ,with ζ a form that is a linear combination each term of which involves at most

n− 1 of the dξi’s and hence at least one dxj . Thus π∗η ∧ τ is, in these coordinates,

a(x)b(x, ξ)dxk ∧ dξn.


We compute

∫Vπ∗η ∧ τ = ∫ a(x)b(x, ξ)dxk ∧ dξn = ∫ ∫ a(x)b(x, ξ)dξdx,

in the last step we replaced the integral of forms by the Lebesgue integral (theusual normalization) and used Fubini. Furthermore, this equals

∫ (a(x)∫ b(x, ξ)dξ)dx = ∫ a(x)c(x)dx != ∫ a(x)dx,

where c(x) ∶= ∫Vx τ and the last equality is 6.2.11. Since the last equation holds

for each compactly supported function a, it follows that c(x) = 1, and this finishesthe proof that τ is indeed a Thom form.

The case of a general base (nonoriented, noncompact) is reduced to this caseby the following trick. We know, by classification of vector bundles, that there isan orientation preserving bundle map f ∶ V → Vn,r, where Vn,r is the tautologi-

cal oriented bundle over the Grassmannian Grn,r of oriented n-planes in Rr, forsome large r. The Grassmannian is a 2-fold cover of the ordinary GrassmannianGrn,r, and because Grn,r is compact, so is the oriented Grassmannian. We haveto argue why the oriented Grassmannian is an orientable manifold (this is not atautology: the tautological vector bundle is not at all the tangent bundle of theGrassmann manifold). But Grn,r is simply connected and hence orientable: theoriented Grassmannian is SO(r)/SO(n) × SO(r − n). The long exact homotopysequence

π1(SO(n) × SO(r − n))→ π1(SO(r))→ π1(Grn,r)→ π0(SO(n) × SO(r − n)) = 0

becomes

(Z/2)2 → Z/2→ π1(Grn,r)→ 0

and the first map is surjective (at least if r ≥ 2, which can be assumed withoutloss of generality). What this argument proves is that each oriented bundle has abundle map f to an oriented vector bundle over a compact oriented base. If σ isa Thom form for the tautological bundle (provided by the first part of the proof),then f∗σ is a Thom form for V .

Theorem 6.2.12. (The Thom isomorphism theorem) Let M be a manifold andπ ∶ V → M be a smooth oriented vector bundle, of rank n. Let τ be a Thom formon V . Then the chain maps

th ∶ A∗(M)→ A∗+ncv (V ); th ∶ A∗c(M)→ A∗+nc (V ),defined by α ↦ π∗α ∧ τ , are quasiisomorphisms.

Proof. This is by the bootstrap lemma. There is not much to say if M = ∅. Eachpoint x ∈ M has a chart neighborhood U ≅ Rm such that V ∣U is trivial. So wehave to show that for the trivial vector bundle on Rn, the theorem holds. Letus do the compactly supported case first. By the computation of H∗

c (Rk), allthat remains to be done is that th ∶ Hc(Rm) → H∗+n

c (Rn) is nonzero. But ifφ = a(x)dx1 ∧ . . . ∧ dxm ∈ Amc (Rm), then

86 JOHANNES EBERT

∫Rm+n

π∗φ ∧ τ = ∫Rm

a(x)∫x×Rnτdx = ∫

Rmφ

by the computation in the proof of Theorem 6.2.10. In the noncompactly supportedcase, we have to show:

Hkcv(Rm ×Rn) = 0 unless k = n.

dimHncv(Rm ×Rn) = 1 and

th ∶H0(Rm)→Hncv(Rm ×Rn) is injective.

We have already seen that integration over x × Rn defines a map Hncv(Rm ×

Rn) → R, and it takes the Thom class to 1, so the third property holds. By thehomotopy invariance of H∗

cv in the base space, we find that

H∗cv(Rm ×Rn) ≅H∗

cv(Rn)which implies the first two properties.The proof of the Mayer-Vietoris and disjoint union property is completely anal-

ogous to the proof of Theorem 6.2.1. The bootstrap lemma applies to conclude theproof.

Corollary 6.2.13. The cohomology class of a Thom form (the Thom class) isuniquely determined by the orientation of V .

Proof. If M is a disjoint union ∐iUi and Vi ∶= V ∣Ui , then H∗cv(V ) ≅ ∏i∈I H

∗cv(Vi).

Therefore, it is enough to check the case when M is connected. By Theorem 6.2.12,Hncv(V ) ≅H0(M) ≅ R. By Stokes theorem, for each x ∈M , we get a homomorphism

Jx ∶Hncv(V )→ R, η ↦ ∫Vx η. A Thom form maps to 1, and therefore Jx is a nonzero

map between 1-dimensional vector spaces, hence an isomorphism. Thus an elementin Hn

cv(V ) is uniquely determined by its integral over Vx.

Lemma 6.2.14. Let p ∶ V → M , q ∶ W → N be two oriented vector bundles, withThom forms τV , τW . Then a Thom form of the product bundle V ×W →M ×N isgiven by τV × τW = p∗τV ∧ q∗τW .

Let f ∶M → N be a smooth map, which is covered by an orientation preserving

bundle map f ∶ V →W . Then f∗τW = τW .

The proof is trivial: you just have to check that the product satisfies the axiomsfor a Thom class, which requires little more than Fubinis theorem. If F is a bundleautomorphism of V , we can talk about the determinant det(Fx). The sign ofdet(Fx) is a locally constant function of x, and we can pull sign(det(F )) back toV ; this is a locally constant function, in other words, an element σ(F ) of H0(V ).It is easy to verify that

(6.2.15) F ∗τV = σ(F )τV .Let V → M be equipped with a bundle metric and let ε ∶ M → (0,∞). Then

we can find a Thom form which has support in DεV ∶= v ∈ V ∣∣v∣ < ε(π(v)), bythe following procedure. For each positive function a ∶M → R, we take the bundleautomorphism ha(v) ∶= 1

a(π(v))v, and by picking a small enough, the form h∗aτ is a

Thom form and has the desired property.

Definition 6.2.16. Let V →M be an oriented vector bundle. The Euler class ofV is ι∗τ ∈Hn(M), where ι ∶M → V is the zero section.


Since two sections are homotopic, one could use any other section instead of thezero section.

The Euler class satisfies some easily verified properties:

Proposition 6.2.17.

(1) The Euler class is natural, i.e. if f ∶ N → M is smooth and V → M anoriented vector bundle, the f∗e(V ) = e(f∗V ).

(2) Reversing the orientation of V reverses the sign of the Euler class.(3) e(V ⊕W ) = e(V ) ∧ e(W ).(4) If V has a section that is nowhere zero, then e(V ) = 0.(5) If the rank of V is odd, then e(V ) = 0.

The first four statements are straightforward to prove, but the last requires anidea. Any vector bundle has the automorphism F = −1. If the rank is odd, then Fis orientation reversing, and thus

e(V ) = ι∗τ = ι∗F ∗F ∗τ1= −ι∗F ∗τ

2= −ι∗τ = −e(V ).The equation 1 is from 6.2.15 and 2 is because F ι = ι.

6.3. Geometric interpretation of the Thom class and the Poincare-Hopftheorem. Let Nn ⊂Mm be a submanifold. We assume that M and N are oriented,which induces an orientation on the normal bundle. We assume that N is compact.Under these circumstances, we get a linear map

`N ∶Hn(M)→ R; ω ↦ ∫Nω.

IfM has finite type, then there exists, by Poincare duality, a unique δ ∈Hm−nc (M)

such that E(δ) = `N , or

∫Mω ∧ δ = ∫

Nω

holds for all closed forms ω on M . If M is the total space of the oriented vectorbundle V → N , then the Thom form has this property. We now present a littlegeometric argument to get rid of the finite type assumption and which gives a nicegeometric interpretation of Poincare duality. The main idea is that each closedsubmanifold of a manifold sits inside the large manifold just as the zero section liesinside a vector bundle. The precise technical ingredient from differential topologythat we need is the tubular neighborhood theorem.

Theorem 6.3.1. (The tubular neighborhood theorem) Let N ⊂ M be a compactsubmanifold. Choose a Riemann metric on M , so that the normal bundle E ofN in M is just the orthogonal complement of TN inside TM ∣N . Then each openneighborhood O of N contains a smaller open neighborhood N ⊂ U ⊂ O such thatthere is a diffeomorphism e ∶ E → U , having the following properties:

(1) The restriction of e to N ⊂ E is the identity.(2) Under the natural splitting (TE)∣N ≅ TN ⊕E, induced by the metric, and

TM ∣N ≅ E ⊕ TN , the differential of e at points of N is the identity.

For further reference, let us note that e is orientation preserving if M and N areoriented and E is equipped with an orientation by the orientation convention.

The idea of the tubular neighborhood theorem is simple, but the details arehighly nontrivial. An unabridged proof can be found in [20], §12. Now let τ ∈

88 JOHANNES EBERT

Am−nc (E) be a Thom form (since N was assumed to be compact, this has compact

support). We let δ ∈ Am−n(M) be the form (e−1)∗τ , extended by zero to all of M .If ω ∈ An, then

∫Mω ∧ δ = ∫

Uω ∧ (e−1)∗τ = ∫

Ee∗ω ∧ τ = ∫

Nω

(use that e is orientation-preserving), by the way the Thom class over a com-pact manifold was constructed. So the form δ represents the functional `N . Notethat this shows that `N therefore lies in the image of E, regardless of finitenessassumptions and therefore does not lie in the outlandish part of the dual space. Ageometric picture shows that the class of δ is supported in a small neighborhood ofN .

Theorem 6.3.2. Let Nn ⊂Mm be a compact oriented submanifold of an orientedmanifold. Then there exists a unique δ ∈ Hm−n

c (M) such that for all ω ∈ Hn(M),we have

(6.3.3) ∫Mω ∧ δ = ∫

Nω.

Rest of the proof. The uniqueness of δ remains to be proven. Let ε ∈ Hm−nc (M)

have the property that

∫Mω ∧ ε = 0

holds for all ω ∈ Hm−n(M). This can be reformulated by saying that for allω ∈ Hm−n(M): D(ω)(ε) = 0, or, by Poincare duality that `(ε) = 0 for all ` ∈Hm−nc (M)∨, in other words, ε = 0.

We call the class δ = δN the Poincare dual to N . It is represented by formsthat are supported in a tubular neighborhood of N , and the support can be madearbitrarily small. The defining equation 6.3.3 is the nonlinear analog of the relation6.2.11. Now we look at the behaviour of the Poincare duals under smooth maps,which can be viewed as a vast generalization of the defining property of the Thomclass 6.2.9. For this, we recall the notion of transversality.

Assume that f ∶ Ll → Mm is a smooth map of oriented manifolds. Assumethat f is transverse to N , f ⋔ N , which means by definition that Tf(TxL) +Tf(x)N = Tf(x)M holds for all x ∈ L, f(x) ∈ N . It follows that K ∶= f−1(N) is asubmanifold of L of the same codimension as N , and that there is a natural bundle

map f ∶ νLK → νMN over f ∣K . An orientation on K is induced by this bundle map andthe orientation convention. The important transversality theorem from differentialtopology states that for each map g ∶ Ll →M , there is a map f ∶ L→M , arbitrarilyclose to g which is transverse to N . For the proof, see [20]. Our goal is:

Theorem 6.3.4. Let f ∶ L → M be a proper map of oriented manifolds which istransverse to the oriented closed submanifold N ⊂ M and let K ∶= f−1(N). LetδN ∈Hm−n

c (M) be the Poincare dual to N . Then f∗δN is the Poincare dual to K.

We begin with a characterization of a form representing δN . Using the tubularmap e ∶ E → U ⊂M , we can and will identify the normal bundle of N in M with aneighborhood of N (also relatively compact).


Definition 6.3.5. Let Nn ⊂Mm be a closed oriented submanifold of the orientedmanifold M . Let N ⊂ U ⊂ E ⊂M be the unit disc bundle in the normal bundle. Amap f ∶ Rm−n →M is a simple cut for N if the following conditions hold:

(1) f−1(N) = 0, f(0) = x0 ∈ N ,(2) f−1(U) ⊂ Rm−n is relatively compact,(3) f ⋔ N (thus f is an immersion near 0),(4) The vector space isomorphism id + T0f ∶ Tx0N ⊕ T0Rm−n → Tx0M is orien-

tation preserving.

For example, the composition Rm−n ≅ Ex → E of an orientation-preserving iso-morphism with the fibre inclusion is a simple cut through x.

Proposition 6.3.6. A closed form δ ∈ A∗c(U) is a Poincare dual to N if and onlyif for each simple cut f to N , the integral ∫Rm−n f

∗δ = 1.

Proof. If the integral is 1 for each simple cut, then the integral ∫Ex δ = 1 for eachx ∈ N . This proves that δ is a Thom form of E, transplanted to M . Vice versa,we have to prove that for each simple cut, the integral is 1. Let f ∶ Rm−n →M bea simple cut through x0 ∈ N . Pick an oriented coordinate chart x for N aroundx0, with x(x0) = 0 and oriented bundle coordinates y for E. Altogether, we get anorientation preserving diffeomorphism (x, y) ∶ W → Rm, sending x0 to 0. By theremarks preceeding Definition 6.2.16, we can rechoose δ to have arbitrarily smallsupport. In particular, we can choose the support so small that if y ∈ Rm−n satisfiesf(y) ∈ supp(δ), then f(y) ∈ W . With these manipulation, we have completelylocalized the situation.

The map f is represented by a map

(f1, f2) ∶ Rm−n → Rn ×Rm−n.

The function f2 has only one zero, namely at zero, and the transversality condi-tion says that 0 is a regular value of f2. Moreover, the orientation assumption meansthat D0f2 has positive determinant. Furthermore, we can assume that f−1

2 (Dl) iscompact and that δ ∈ Am−n(Rn×Rm−n) is a closed form with support in Rn×Dm−n,such that ∫x×Rm−n δ = 1 for all x ∈ Rn.

We have to evaluate the integral ∫Rm−n(f1, f2)∗δ, which is difficult without atrick. The trick is to consider the family ft = (tf1, f2), t ∈ [0,1]. By our assumptionon the support of δ and f2, the smooth family τt ∶= f∗t τ has compact support forall t. Therefore the integral ∫Rm−n τt does not depend on t. We want to knowthe value for t = 1. But the computation is easier for t = 0. Namely, we have tocompute ∫Rm−n f

∗2 δ. By further shrinking the support of δ, we can arrange that f2

is an orientation-preserving diffeomorphism f−1(supp(δ)) → supp(δ). Therefore,the integral is 1.

Proof of Theorem 6.3.4. Let K ⊂ V ⊂ L be a tubular neighborhood. Fix y ∈ K, letg ∶ Rm−n ≅ Vy ⊂ V be the composition of an orientation-preserving diffeomorphismwith the fibre inclusion. View Rm−n as the interior of Dm−n and if V is chosensufficiently small, then g extends to a smooth map g ∶ Dm−n → V ⊂ L. The mapf g ∶ Rm−n ⊂ V → M is a cut through N , except that the second condition ofDefinition 6.3.5 might fail, which is corrected as follows.

Pick a Riemann metric on M ; there is an ε > 0 such that dist(f(z),N) ≥ 2ε forall z ∈ Sm−n−1. Pick a tubular neighborhood N ⊂ U ⊂ M that is contained in the

90 JOHANNES EBERT

ε-neighborhood on N in M and choose the form δ ∈ Am−nc (M) to be supported in

this tubular neighborhood. Then f g is a cut, and to finish the proof, we compute

∫Rm−n

g∗f∗δ = ∫Rm−n

(f g)∗δ.

Since fg is a cut, the right-hand side is 1.

Theorem 6.3.4 has some interesting applications.

Theorem 6.3.7. Let Mm be a closed oriented manifold and π ∶ V → M be anoriented vector bundle of rank n. Let s be a section of V which is transverse tothe zero section and let Zm−n ∶= s−1(0) ⊂M be the zero set. This submanifold hasan induced orientation, because νMZ ≅ V ∣Z . Then the Euler class e(V ) ∈ Hn(M) isPoincare dual to Z.

Proof. Let s0 be the zero section and let τ be a Thom class for V . Then e(V ) =s∗0τ = s∗τ . Since τ is the Poincar’e dual of M in V , it follows that s∗τ is thePoincare dual of Z in M .

The case m = n is of particular interest. In this case, Z is a finite set Z =x1, . . . , xr; each point xi comes equipped with a sign εi ∈ ±1 that determines itsorientation. By Theorem 6.3.7, we compute

∫Me(V ) = ∫

MδZ

6.3.2= ∫Z

1 =r

∑i=1

εi.

The signs εi are called the local indices of the section s and denoted Ixis. Wehave proven

Theorem 6.3.8. (Poincare-Hopf theorem) If V → M is an oriented rank n vec-tor bundle over an oriented closed n-manifold and s a cross-section of V that istransverse to 0, then ∫M e(V ) = ∑s(x) Ixs.

Another interesting application is:

Theorem 6.3.9. Let Mm be an oriented manifold of dimension and let Ll,Nn ⊂Mbe two closed oriented submanifolds. Assume that L and N intersect transversally,in other words ι ⋔ N , where ι ∶ L→M is the inclusion. Then δMN ∧δML is a Poincaredual to K = L ∩N .

Proof. Let ω be a closed form on M . Then

∫Kω∣K = ∫

Lω∣L ∧ δLK = ∫

L(ω ∧ δMN )∣L = ∫

Mω ∧ δMN ∧ δML .

We are sloppy about signs here; ultimately, we are interested in even-dimensionalmanifolds only, where all sign questions disappear.

Exercise 6.3.10. Determine the signs in the previous theorem.

We now turn to a fundamental computation. Let H → CPn be the dual tothe tautological line bundle. Here, it is useful to do the computation in the mostinvariant way. Let V be a complex vector space of dimension n + 1. Why do weuse the dual? Let f ∈ V ∗ be a linear form on V . By restriction, f induces a linearform on each line ` ∈ PV , in other words, a section sf of the bundle L∗, which is


a holomorphic section. One could try the dual thing, but the only way to producta section of L out of a vector v ∈ V is by projecting v to `. This is fine, but theprojection involves conjugates which is why the induced section is not holomorphic.In fact, L does not have any nonzero holomorphic section.

The zero set of sf is the set of all lines such that f ∣` = 0, or the projective spaceof ker(f). Let us assume that f ≠ 0; then it can be shown without difficulty thatsf is transverse to the zero section.

Theorem 6.3.11. Let H → CPn be the dual of the tautological bundle. Then

∫CPn

e(H)n = 1.

Proof. If n = 1, then the section sf (f ≠ 0) has a unique zero, since ker(f) is 1-dimensional. Because the section is holomorphic, the local index must be +1, not−1, and the claim follows from the Poincare-Hopf theorem 6.3.8. For higher n,we use that e(H)n = e(H⊕n). Take linearly independent f1, . . . , fn ∈ V ∗. Theyinduce a holomorphic section s = (sf1 , . . . , sfn) which has a unique zero, namelythe line ∩ni=1 ker(fi). Again, the local index must be +1 by holomorphicity, andPoincare-Hopf finishes the proof.

Now we can easily compute the cohomology ring of CPn.

Theorem 6.3.12. Put x ∶= e(H). Then

H∗(CPn) = R[x]/(xn+1)and ∫CPn x

n = 1.

Proof. We have just computed the integral, which shows that xk ≠ 0 for all k ≤n. It remains to compute the dimensions of Hi(CPn). For that, we use thatCPn ∖CPn−1 ≅ Cn and the sequence 6.1.4:

H∗(CPn) // H∗(CPn−1)

[+1]xxH∗c (Cn)

ff

which gives an exact sequence

0 =H2n−1(CPn−1)→ R ≅H2nc (Cn)→H2n(CPn)→H2n(CPn−1) = 0

and that the restriction maps are isomorphisms

Hk(CPn) ≅Hk(CPn−1)for k ≤ 2n − 1. Together, this makes an inductive proof, starting from the casen = 0.

6.4. The topological Gauß-Bonnet theorem.

Theorem 6.4.1. Let Mn be a closed oriented manifold. Then

∫Me(TM) = χ(M).

As a corollary, we obtain the first nontrivial instance of the Atiyah-Singer indextheorem:

92 JOHANNES EBERT

Corollary 6.4.2. Let M be a closed oriented Riemann manifold and d + d∗ ∶Aev(M)→ Aodd(M) be the Euler characteristic operator. Then

ind(d + d∗) = ∫Me(TM).

Lemma 6.4.3. Let M be a manifold and ∆ ∶M →M ×M be the diagonal embed-ding, with image ∆(M). Then the normal bundle of ∆(M) in M ×M is naturallyisomorphic to the tangent bundle TM ; and the isomorphism can be chosen to pre-serve orientations.

Let U be a tubular neighborhood of ∆(M) and let V be the normal bundle of∆(M). Let τ be a Thom form of TM and let ρ ∈ An(M ×M) be the result ofgrafting the Thom form τ into M ×M (in other words, ρ is the Poincare dual to∆(M)). Note that

e(TM) = ∆∗e(V ) = ∆∗ι∗ρ

where ι is the inclusion. Thus e(TM) = ∆∗ρ. Now let α be a homogeneousbasis of H∗(M) and let β# be the dual basis, i.e.

∫Mα# ∧ β = δα,β .

By the Kunneth theorem, α# ×β is a basis for H∗(M ×M). There are uniquecα,β ∈ R with

ρ = ∑α,β

cα,βα × β#.

For two basis elements, γ, ε, we compute ∫M×M(γ# × ε)∧ ρ in two ways. First ofall

∫M×M

(γ# × ε) ∧ ρ 1= ∫M

∆∗(γ# × ε) = ∫Mγ# ∧ ε = δγ,ε.

In the first equation, we used the fact that the Thom class is Poincare dual tothe diagonal. The other way to compute is

(6.4.4) ∫M×M

(γ# × ε) ∧ ρ = ∑α,β

cα,β ∫M×M

(γ# × ε) ∧ (α × β#).

But

∫M×M

(γ# × ε) ∧ (α × β#) = (−1)∣ε∣∣α∣ ∫M×M

(γ# ∧ α) × (ε ∧ β#) =

= (−1)∣ε∣∣α∣ ∫M

(γ# ∧ α)∫M

(ε ∧ β#) = (−1)∣ε∣∣α∣δγ,α(−1)(n−∣β)∣ε∣δβ,εand therefore

δγ,ε = (6.4.4) = cγ,ε(−1)∣ε∣∣γ∣(−1)(n−∣ε)∣ε∣ = (−1)n∣γ∣.In other words

ρ =∑α

(−1)n∣α∣α × α#

and therefore


∫Me(TM) = ∫

M∆∗ρ =∑

α

(−1)n∣α∣ ∫Mα∧α# =∑

α

(−1)n∣α∣∣α∣(n−∣α∣)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

=(−1)∣α∣

∫Mα# ∧ α

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=1

= χ(M).

6.5. The Gysin map and the splitting principle for complex vector bun-dles.

Definition 6.5.1. Let f ∶Mn+d → Nn be a map between closed oriented manifolds(we allow d to be negative). The Gysin map f! ∶ Hk(M) → Hk−d(N) is defined tobe the composition

Hk(M) DM→ Hn+d−k(M)∗(f∗)∗→ Hn+d−k(N)∗

D−1N→ Hk−d(N).

The two stars in the symbol (f∗)∗ mean two different things: the inner ∗ isthe induced map on cohomology, and the outer ∗ is the dual in the sense of linearalgebra. Some remarks on terminology: often the Gysin map is called umkehr map(also in the English literature). I want to warn against two misnamings that occurquite often. The first misnaming is ”pushforward”. This is casual terminology, andI do not want to advertise this. The second misnaming is ”transfer”, and using thisword for the Gysin map is an outright mistake - the true use of the word ”transfer”is for something closely related, but different.

Let us unwind the definition of the Gysin map. Let ω ∈ Hk(M) and η ∈Hn+d−k(N). Then we compute (DNf!(ω))(η) = ∫N f!(ω) ∧ η by the definitionof DN . On the other hand

(DNf!(ω))(η) = ((f∗)∗DM(ω))(η) =DM(ω)(f∗η) = ∫Mω ∧ f∗η.

Thus we arrive at the equation

(6.5.2) ∫Nf!(ω) ∧ η = ∫

Mω ∧ f∗η

which characterizes f! and can be expressed by saying that f! is adjoint to f∗ withrespect to the duality pairing. We will use equation 6.5.2 to derive all properties ofthe Gysin map. First

∫Nf!(ω ∧ f∗ζ) ∧ η = ∫

Mω ∧ f∗ζ ∧ f∗η = ∫

Mω ∧ f∗(ζ ∧ η) = ∫

Nf! ∧ ζ ∧ η.

Since this holds for all η, we find - using duality - that

(6.5.3) f!(ω ∧ f∗ζ) = f!(ω) ∧ ζ.For another consequence, consider the constant map f ∶ M → ∗. Then f!(ω) =

∫M ω. Also

(f g)! = f! g!

is an immediate consequence of the functoriality of cohomology.

Proposition 6.5.4. Let f ∶Mn+d → Nn be a smooth map of closed oriented man-ifolds, let z ∈ N be a regular value of f . Assume that N is connected and letω ∈Hd(M). Then f!(ω) = ∫f−1(z) ω ∈ R =H0(N).

94 JOHANNES EBERT

Proof. Let τ ∈ Hn(N) be the Poincare dual to z ⊂ N (this is just a class with

∫N τ = 1). By Theorem 6.3.4, f∗τ is the Poincare dual to f−1(z) in M . Then

∫Nf!(ω) ∧ τ

6.5.2= ∫Mω ∧ f∗τ 6.3.2= ∫

f−1(z)ω.

A nice situation appears if all z ∈ N are regular values, in other words, if f isa submersion. A not so hard theorem from differential topology, the Ehresmannfibration lemma, says that a proper submersion is a fibre bundle. In this situation,one can give an explicit differential form representative of f!(ω), obtained by inte-gration over the fibres. We do not need to consider this refinement of the definitionof the Gysin homomorphism here.

If f ∶ M → N is a submersion, we denote by TvM ∶= ker(df) the kernel of thedifferential of f . This is a vector bundle by Lemma 5.1.6 and called the verticaltangent bundle; because it consists of all tangent vectors that are tangent to thefibres of f . There is a vector bundle splitting

(6.5.5) TM ≅ f∗TN ⊕ TvE.

If M and N are both oriented, then TvE acquires an orientation. The topologicalGauss-Bonnet theorem has two interesting consequences.

Definition 6.5.6. Let f ∶M → N be a proper submersion of closed oriented man-ifolds. The transfer trff ∶H∗(M)→H∗(N) is the map trff(ω) ∶= f!(e(TvM)ω).

Theorem 6.5.7. Let f ∶ M → N be a proper submersion of closed oriented man-ifolds. Assume that M is connected and that the Euler characteristic χ(F ) of thefibres F ∶= f−1(x) is nonzero. Then f∗ ∶H∗(N)→H∗(M) is injective.

Proof. We calculate

trff(f∗ω) = f!(e(TvE)f∗ω) 6.5.3= f!(e(TvE))ω 6.5.4= χ(F )ω.Since χ(F ) ≠ 0, this implies that f∗ is injective.

Theorem 6.5.8. Let f ∶ Mn+d → Nn be a proper submersion of closed orientedmanifolds, with F ∶= f−1(z). Then χ(M) = χ(N)χ(F ).

Proof. This is a straightforward consequence of the topological Gauss-Bonnet the-orem:

χ(M) = ∫Me(TM) = ∫

Mf∗e(TN) ∧ e(TvE) = (−1)dn ∫

Me(TvE) ∧ f∗e(TN) =

= (−1)nd ∫Nf!(e(TvE)) ∧ e(TN) = (−1)ndχ(F )∫

Ne(TN) = (−1)ndχ(F )χ(N).

If nd is odd, then both F and N are odd-dimensional and thus have zero Eulernumbers, and so does M by the above equation, and the sign does not matter. Inall other cases, χ(M) = χ(F )χ(N), as claimed.


6.6. Literature. The basic material on de Rham cohomology theory is standardand there are many elementary expositions, as [42], [53]. The de Rham theorem(which is proven here) is of great conceptual importance, even if we do not use it.A slick proof is given in Bredon’s book [19]. Everything in this chapter that goesbeyond the basic properties owes much to the exposition by Bott and Tu [17]. Ialso consulted the book by Madsen and Tornehave [49]. There is a Bourbakian texton the subject: the three-volume treatise [28], [29], [30].

96 JOHANNES EBERT

7. Connections, curvature and the Chern-Weil construction

Now we develop the local theory of characteristic classes.

7.1. Covariant derivatives.

Definition 7.1.1. Let V be a vector bundle on a manifold M . A covariant de-rivative alias connection is a map ∇ ∶ A0(M ;V ) → A1(M ;V ) such that ∇(fs) =df ⊗ s + f∇s.

We will use the two terms interchangingly; later, we will describe the sameobject in a different language, and then we distinguish the names properly. We seethat [∇, f] = df ⋅. In other words, a covariant derivative is a first order differentialoperator whose principal symbol is given by smb∇(ξ) = iξ for all cotangent vectorsξ. Therefore, covariant derivatives exist on any vector bundle (Proposition 2.2.19).We want some more concrete examples.

Proposition 7.1.2. The exterior derivative A0 → A1 is a connection. Let V ⊂M × Rn and p the orthogonal projection onto V . Then ∇ ∶= pd is a covariantderivative.

The proof is trivial. On a Riemann manifold, there is a special covariant deriv-ative on the tangent bundle.

Definition 7.1.3. Let V →M be a Riemannian vector bundle. A covariant deriv-ative ∇ is called metric if X⟨s, t⟩ = ⟨∇Xs, t⟩ + ⟨s,∇Xt⟩ holds for all vector fields Xand all sections s, t.

Theorem 7.1.4. (The fundamental lemma of Riemannian geometry) On any Rie-mann manifold, there is a unique connection on TM , which is metric and torsion-free, ∇XY −∇YX = [X,Y ]. This connection is also called Levi-Civita connection.

The proof can be found in any textbook on Riemann geometry.

Lemma 7.1.5. Let V → M be a vector bundle and ∇ a connection on V . Thenthere is a unique sequence of linear maps (differential operators)

A0(M ;V ) ∇→ A1(M ;V ) ∇→ A2(M ;V ) ∇→ . . .

such that ∇ coincides with the connection for p = 0 and such that ∇(ω ∧ s) =dω ∧ s + (−1)∣ω∣ω ∧∇s holds.

Proof. Locally, each s ∈ Ap(M ;V ) can be written as a linear combination of termsof the form ω⊗ t, where ω is a scalar-valued form and t a section of V . The productrule prescribes the value of ∇ on these elements. This shows uniqueness.

Choosing a local frame e1, . . . , er of V , each section can uniquely written ass = ∑i ωi ⊗ ei, for forms ωi. We set

∇s =∑i

dωi ⊗ ei + ωi ∧∇ei.

This has the desired property locally, and uniqueness proves that it is coordinateindependent.

Note the similarity to the definition of the exterior derivative. But we have tosacrifice the condition ∇2 = 0, for a very substantial reason, as we shall see.


Proposition 7.1.6. There exists a unique 2-form Ω with values in End(V ) suchthat ∇2 = Ω. This form is called the curvature.

Proof. First we prove that ∇2 is of order 0. This is because

∇∇(fs) = ∇(df ∧ s) +∇(f∇s) = −df ∧∇s + df ∧∇s + f∇2s.

More generally, if ω is a form, then

∇∇(ω ∧ s) = ∇(dω ∧ s) + (−1)p∇(ω∇s) = ω∇2s.

Therefore, ∇2 commutes with multiplication by forms and thus it is determinedby Ω.

7.2. The first Chern class - the baby case of Chern-Weil theory. Let usfigure out what the curvature look like for a complex line bundle L → M . Let sbe a local section of L without zeroes, in other words a local basis. As ∇s is anL-valued 1-form, we can write it as ∇s = ω ∧ s, for a unique complex valued 1-formω. But then

∇2s = ∇(ω ∧ s) = dω ∧ s − ω ∧∇s = (dω) ∧ s − ω ∧ ωs = (dω) ∧ s.Here we used in an essential way that we talked about a line bundle. The

relevant property is that L has abelian structure group, as we will see. We haveshown that the curvature form is just Ω = dω. That it is a scalar valued 2-formshould not come as a surprise: the endomorphism bundle of a line bundle is trivial,in a canonical way (the identity endomorphism is a global section without zeroes).Let us make some remarks. We know that the form Ω is a globally defined 2-form,by Proposition 7.1.6! The formula Ω = dω seems to suggest that Ω is an exactform, but this is fallacious: the form ω depends on the choice of s and therefore, itdoes exist only locally. But the formula Ω = dω does give important information -namely that Ω is closed, which is by definition a local property of a form. So we geta cohomology class [Ω] ∈ H2(M). If L is trivial, then there exists a global sections without zeroes, and thus the form ω has a global meaning, which tells us at oncethat Ω is exact and [Ω] = 0.

On the other hand, the trivial bundle M × C admits a connection whose cur-vature is zero, namely the exterior derivative! These observations might lead usto the suspicion that the cohomology class of the curvature form contains relevantinformation about the global structure of the bundle. We now prove that is indeedcorrect.

Definition 7.2.1. Let L→M be a complex line bundle and ∇ a connection on Lwith curvature form Ω. The first Chern class of L is the class

c1(L) ∶= −1

2πi[Ω].

Lemma 7.2.2. The cohomology class of Ω is independent of the choice of theconnection.

Proof. Indeed, the difference ∇1 − ∇0 of two connections on L is an operator oforder 0 and hence given by a (complex-linear) vector bundle homomorphism L →T ∗M ⊗R L or equivalently by a 1-form with values in the endomorphism bundle

98 JOHANNES EBERT

End(L). As End(L) is trivial, this is just a 1-form α ∈ A1(M,C). Hence ∇1 = ∇0+αfor a globally defined 1-form α. With respect to a local section s, this shows thatω1 = ω0 + α (locally) and hence Ω1 = Ω0 + dα globally.

One can (and we will) show that if f ∶M → N is a smooth map, then f∗c1(L) =c1(f∗L), but this won’t be simpler than the general argument given below.

Example 7.2.3. Consider the tautological line bundle L → CP1. Recall thatL = (`, v) ∈ CP1∣v ∈ `. Let U ⊂ CP1 be the set of all [1 ∶ z] ∈ CP1; this is thecomplement of a point. We will now compute the projection connection ∇ and itscurvature. There is a complex chart C → U , z ↦ [1 ∶ z]. Over U , we have thesection s ∶ U → L∣U , which expressed in these coordinates is given by

s(z) ∶= ([1 ∶ z], (1, z)).

In other words, the section s is given by the vector valued function, also denoted

s(z) = (1z). The projection operator is

p(z) = 1

∥s(z)∥(1z)(1 z) = 1

1 + ∣z∣2(1 zz ∣z∣2) .

With these formulae, we compute

∇s = p(ds) = 1

1 + ∣z∣2(1 zz ∣z∣2)( 0

dz) = 1

1 + ∣z∣2( zdzzzdz

) = zdz

1 + ∣z∣2(1z) = ωs

with

ω = zdz

1 + ∣z∣2.

The curvature is the exterior derivative of this form, but we do not need tocompute the derivative. Let us show that [Ω] and hence c1(L) is nontrivial, bycomputing ∫CP1 c1(L). Since the complement of the coordinate patch U is a point,it has measure zero, and we compute

−∫CP1

c1(L) =1

2πi∫Cdω = 1

2πilimr→∞∫∣z∣≤r

dω = 1

2πilimr→∞∫∣z∣=r

ω

by Stokes theorem. But for ∣z∣ = r, we have z = r2z−1 and hence

1

2πilimr→∞∫∣z∣=r

ω = 1

2πilimr→∞

r2

1 + r2 ∫∣z∣=rz−1dz = 1.

The minus sign has ”historical” reasons, the 2πi factor emphasizes the integralstructure of cohomology! The ”historical” reason is to line up with the Euler class.

Theorem 7.2.4. Let L→M be a complex line bundle. Then c1(L) = e(L).

The proof is a prelude to the proof of the Gauß-Bonnet-Chern theorem, and is agood introduction to the techniques involved in the theory of characteristic classes.First, we use that any complex line bundle has a bundle map to L → CPn, thetautological line bundle. The naturality of the Euler class, together with the notyet proven naturality of the first Chern class, shows that it is enough to considerthe tautological line bundle.


Now the dual of a complex vector bundle is isomorphic to the complex-conjugatebundle. The effect on the Euler class is that orientation is changed by (−1)k, k therank. Therefore, the Euler class of L is −1, the same as c1(L).

The computation for the integral of the first Chern class on CP1 proves that itis equal to the Euler class. The case of higher dimensional projective spaces followsbecause H2(CPn)→H2(CP1) is an isomorphism!

However, not all vector bundles are line bundles, and we now study connectionsand curvature in more detail for higher rank bundles. It is the failure of commuta-tivity of the Lie groups GLn that makes this more difficult.

7.3. The coordinate description of a connection. In the same way as principalbundles gave us more flexibility when dealing with bundles, we will gain flexibilitywith connections by introducing a new concept a connection on aG-principal bundlewhen G is a Lie group. Just as principal bundles is a notion to keep track ina systematic way of all trivializations, the notion of a connection on a principalbundle arises from a systematic study of the way the connection is written whencoordinates are chosen.

Let V →M be a vector bundle and ∇ be a connection on V . Let U ⊂M be openand s a local section of the frame bundle, defined over U . We consider s as a mapU ×Rn → V ∣U (a bundle isomorphism). We obtain isomorphisms

φs ∶ Ap(U ;Rn)→ Ap(U ;V ).These isomorphisms have the following explicit description. Let ei be the ith

unit vector in Rn. Recall that for x ∈ U , s(x) gives a vector space isomorphismRn → Vx, and we let si(x) ∈ Vx be the image of ei. Of course, we obtain continuoussections si of V ∣U , and this is another way of describing the local frame. For p = 0,the map φs sends a function a = (a1, . . . , an) to the section ∑i aisi. The same istrue for p-forms, i.e. if ai is a p-form. We obtain a commutative diagram

(7.3.1) Ap(U,Rn)φs //

?

Ap(U ;V )

∇

Ap+1(U ;Rn)φs // Ap+1(U ;V );

the left vertical map is defined so that the diagram commutes (this is possiblesince the horizontal maps are isomorphisms). Let us describe the map ”?”. Thereexists uniquely determined forms

θij ∈ A1(U) such that ∇sj =∑i

θijsi(sic!).

We summarize them in the matrix θ = θs = (θij)i,j=1,...n. Now we follow a tupleof p-forms a = (a1, . . . , an) in the diagram 7.3.1. By φs, it is sent to ∑i aisi and

∇∑i

aisi =n

∑i=1

dai ⊗ si + (−1)p∑j,i

ajθijsi

With these notations, we can write

∇φs(a) =∑i

dai ⊗ si + (−1)p∑j

∑i

ajθijsi = φs(da) + φs(θa),

100 JOHANNES EBERT

the (−1)p factor is absorbed because ai is a scalar-valued p-form and the entriesof θ are 1-forms and we interchanged the order of multiplication.

Thus the connection can be written in local coordinates s as d + θ, for a 1-formθ ∈ A1(U ;gln(R)). Here gln(R) is the Lie algebra of GLn(R); which as a vectorspace is the same as Matn,n(R). Not yet, but soon it will become clear why we usethe Lie algebra notation here. The curvature is easily computes in this formalismas

Ωsa = (d+θ)(d+θ)a = (d+θ)(da+θa) = dda+dθa−θ∧da+θ∧dθ+θ∧θa = (dθ+θ∧θ)a,the notation Ωs indicates that the matrix-valued 2-form Ωs ∈ A2(U,gln(R))

depends on s. As a side remark, this gives another proof that the curvature is atensor field. Finally, we remark that θ ∧ θ is in general not zero, because the ringMatn,n(R) is not commutative.

The next step is to figure out how the forms θs and Ωs change if the local frames changes. Let g ∶ U → GLn(R) be a smooth function and s a local frame. Weobtain a new local frame sg, with components given by

(sg)i = (∑j

gj1sj , . . . ,∑j

gj1sj).

The functions / forms ai are changed to g−1a (matrix multiplication).

The map φsg is the composition φs(g . . . ), where (g) is the map given by matrixmultiplication. To find out the change-of-frame formula, we look at the diagram

C∞(U,Rn)

d+θsg

g⋅ // C∞(U,Rn)φs //

d+θs

A0(U,V )

∇

A1(U,Rn)g⋅ // A1(U,Rn)

φs // A1(U,V );the horizontal compositions are the maps φsg. From that, one derives the formula

(7.3.2) θsg = g−1dg + g−1θsg.

The curvature form transforms as

Ωsg = dθsg + θsg ∧ θsg = d(g−1dg) + d(g−1θsg) + (g−1dg + g−1θsg) ∧ (g−1dg + g−1θsg) == d(g−1dg) + d(g−1θsg) + g−1dgg−1dg + g−1dgg−1θg + g−1θdg + g−1θ ∧ θg =

= d(g−1θsg) + g−1dgg−1θg + g−1θdg + g−1θ ∧ θg == g−1dgg−1θsg + g−1dθg − g−1θdg + g−1dgg−1θg + g−1θdg + g−1θ ∧ θg =

= g−1(dθ + θ ∧ θ)g.Let us summarize the calculations so far:

Proposition 7.3.3. There is a bijection between connections on V and rules thatassign a form θs ∈ A1(U,Matn,n(Rn)) to a local frame s, such that θsg = g−1dg +g−1θsg, for each change-of-frame function g ∶ U → GLn(R). The curvature is givenby Ωs = dθs + θs ∧ θs, and the change-of-frame formula is Ωsg = g−1Ωsg. Any sucha rule defines a connection; in a frame s it is ∇ = d + θs.


This description is not very practical yet; we want to package the informationof the ”rule” in a single 1-form θ ∈ A1(Fr(V );gln(R)), such that θs ∶= s∗θ. And wewant to talk about other Lie groups than GLn(R).

7.4. The very basics of Lie theory. We need to pause a little and introducesome basic constructions of Lie theory, beyond the mere definitions.

Definition 7.4.1. Let G be a Lie group. The Lie algebra of G is the vector spaceT1G = g = Lie(G).

If V is a real vector space, then gl(V ) = End(V ).

Definition 7.4.2. A Lie algebra over a field F of characteristic ≠ 2 is a F-vectorspace g, together with a bilinear map g × g→ g, (X,Y )↦ [X,Y ] such that

(1) [X,Y ] = −[Y,X] and(2) [[X,Y ], Z] + [[Y,Z],X] + [[Z,X], Y ] (the Jacobi identity)

hold.

How is the structure of a Lie algebra on T1G defined? Each g ∈ G defines asmooth map Cg ∶ G→ G, h↦ ghg−1, and Cg(1) = 1.

Definition 7.4.3. Let G be a Lie group. The adjoint representation of G on g isdefined by Ad(g)X ∶=D1Cg(X).

It is easy to see that

Ad(gh) = Ad(g)Ad(h)and that Ad ∶ G→ GL(g) is a smooth group homomorphism.

Definition 7.4.4. Let ad ∶ g → gl(g) be the derivative of Ad at the identity. Wedefine, for X,Y ∈ g: [X,Y ] ∶= ad(X)(Y ).

It is not a complete triviality to prove:

Theorem 7.4.5. (Lie’s first theorem) g, equipped with [, ] is a Lie algebra. Ifφ ∶ G→H is a homomorphism of Lie groups, then D1φ is a homomorphism of Liealgebras.

This can be found in basic texts on Lie theory, e.g. [22], [26].

Example 7.4.6. Let G = GLn(R). Then gln(R) is the space of n × n-matrices.The conjugation map Cg(h) ∶= ghg−1. To compute the adjoint representation, letX ∈ gln(R) and g ∈ GLn(R). Then t ↦ exp(tX) is a curve through 1 with tangentvector X, and

Ad(g)X = d

dt∣t=0g exp(tX)g−1 = gXg−1.

Moreover

ad(Y )(X) = d

dt∣t=0 exp(tY )X exp(−tY ) = Y X −XY.

Definition 7.4.7. A representation of a Lie group G on the vector space V is asmooth group homomorphism φ ∶ G→ GL(V ). A representation of a Lie algebra isa Lie algebra homomorphism ϕ ∶ g→ gl(V ).

102 JOHANNES EBERT

If a Lie group representation φ ∶ G → GL(V ) is given, we obtain a Lie algebrarepresentation as the derivative of φ, at the identity. There are some ways toproduce new representations of Lie groups/algebras out of old ones. We denote theaction by g ∈ G or X ∈ g.

Examples 7.4.8. (1) The trivial representations: g ⋅ v ∶= v, X ⋅ v = 0.(2) Direct sums of representations are representations (in both cases).(3) If V is a representation, then the dual space has the following representa-

tion: g ⋅ ` ∶= ` g−1, X ⋅ ` ∶= −` X.(4) If V and W are representations, the tensor product V ⊗W is a representa-

tion: g(v ⊗w) ∶= gv ⊗ gw, X(v ⊗w) ∶=Xv ⊗w + v ⊗Xw.(5) Hom(V,W ) is a representation: g ⋅ f ∶= g f g−1, X ⋅ f ∶=X f − f X.

Lemma 7.4.9. Let φ ∶ G → GL(V ) be a representation and ϕ ∶ g → gl(V ) be theinduced Lie algebra representation. Then ϕ is G-equivariant, in other words for allg ∈ G and x ∈ g:

φ(g)ϕ(X)φ(g−1) = ϕ(Ad(g)X) ∈ gl(V ).

Proof. Let ct ∶ (−ε, ε)→ G be a curve with c0 = 1 and c0 =X. Then

φ(g)ϕ(X)φ(g−1) = φ(g) ddt

∣t=0φ(ct)φ(g−1) = d

dt∣t=0φ(gctg−1) =

= d

dt∣t=0φ(Cg(ct)) = ϕ(

d

dt∣t=0Cgct) = ϕ(Ad(g)X).

If M is a manifold and g a Lie algebra, we can talk about the space Ap(M ;g) ofp-forms with values in g. One can combine the wedge product and the Lie bracket:

[; ] ∶ Ap(M ;g)⊗Aq(M ;g) ∧→ Ap+q(M ;g⊗ g)[,]→ Ap+q(M,g);

more concretely, if ω, η are real valued forms and X,Y ∈ g, then [ω⊗X,η⊗Y ] ∶=ω ∧ η ⊗ [X,Y ].

Example 7.4.10. Let g = gl(V ) and X,Y ∈ gl(V ), ω ∈ Ap(M), η ∈ Aq(M). Then

[ω ⊗X,η ⊗ Y ] = ω ∧ η ⊗XY − ω ∧ η ⊗ Y X = (ω ⊗X) ∧ (η ⊗ Y ) − (−1)pqη ∧ ω ⊗ Y X =(ω ⊗X) ∧ (η ⊗ Y ) + (−1)pq+1(η ⊗ Y ) ∧ (ω ⊗X).

In other words, if ω ∈ Ap(M,gl(V )) and η ∈ Aq(M,gl(V )), we find that

(7.4.11) [ω, η] = ω ∧ η − (−1)pqη ∧ ω,in particular, for ω ∈ A1(M ;g):

ω ∧ ω = 1

2[ω,ω].

In general, one can prove easily that Ap(M,g) has the structure of a differentialgraded Lie algebra:

Proposition 7.4.12. Assume that ω ∈ Ap(M ;g), η ∈ Aq(M ;g), and ζ ∈ Ar(M ;g).Then


(1) d[ω, η] = [dω, η] + (−1)p[ω, dη],(2) [ω, η] = (−1)pq+1[η,ω],(3) (−1)pr[[ω, η], ζ] + (−1)qp[[η, ζ], ω] + (−1)rq[[ζ, ω], η] = 0.

A Lie algebra homomorphism ϕ ∶ g → h induces, in an obvious manner, a mapϕ∗ ∶ A∗(M,g) → A∗(M,h). Smooth maps g ∶ M → G act on g-valued differentialforms by the adjoint representation. I.e., if ω ∈ Ap(M,g), then Ad(g)ω is the formthat takes tangent vectors X1, . . . ,Xp ∈ TxM to

Ad(g(x))(ω(X1, . . . ,Xp)).The mother of all Lie algebra valued forms is a canonical 1-form that exists on

every Lie group.

Definition 7.4.13. Let G be a Lie group and π ∶ TG → G be the tangent bundle.By Rg, Lg, we denote the left and right translations by g ∈ G. The maps TG→ G×g,v ↦ (π(v), Lπ(v)−1∗v) and G × g → TG, (g, x) ↦ Lg∗x are two mutually inverse

bundle isomorphisms. The 1-form ωG ∈ A1(G;g), v ↦ Lπ(v)−1∗v is called Maurer-Cartan-form.

Example 7.4.14. LetG = GLn(R). Then there is the trivialization TG = GLn(R)×Matn,n(R), since GLn(R) ⊂ Matn,n(R) is an open subset. The map v ↦ Lπ(v)−1∗vbecomes in this trivialization

(g,X)↦ g−1X,

because the left-translation map is ”linear”. So we can write the Maurer-Cartanform ωGLn(R) = g−1dg.

The Maurer-Cartan form has some useful properties.

Proposition 7.4.15. Let G and H be Lie groups and φ ∶ G → H be a homomor-phism with derivative ϕ ∶ g → h. Let µ ∶ G ×G → G be the multiplication. Then thefollowing statements are true:

(1) For all g ∈ G: L∗gωG = ωG; R∗gωG = Ad(g−1) ωG.

(2) φ∗ωH = ϕ∗ωG.(3) µ∗ωG = p∗2ωG +Ad(p−1

2 )p∗1ωG, where pi ∶ G×G→ G are the two projections.(4) (structural equation) dωG + 1

2[ωG, ωG] = 0.

(5) Let η ∈ Ap(M) and g ∶M → G. Then

d(Ad(g−1)η) = −g∗ωG ∧Ad(g−1)(η) +Ad(g−1)dη + (−1)pAd(g−1)ηg∗ωG.

We will only give the proof in the special case when G is a linear group, i.e. ifG ⊂ GLn(R). This is not a serious restriction for our purposes; we will only considerlinear groups, but simplifies the proof, because the objects are easier to grasp.

Proof. Ad 1) and 2): these follow immediately from the definitions, and for arbi-trary Lie groups.

Ad 3) This is the computation

(gh)−1d(gh) = (gh)−1gdh+(gh)−1(dg)h = h−1g−1gdh+h−1g−1(dg)h = h−1dh+h−1(g−1dg)h,together with the formulae for the adjoint representation and the Maurer-Cartanform.

104 JOHANNES EBERT

Ad 4) This is the computation

d(g−1dg) + 1

2[g−1dg, g−1dg] 7.4.11= −g−1dgg−1 ∧ dg + g−1dg ∧ g−1dg = 0.

Ad 5)

d(Ad(g−1)η) = d(g−1ηg) = −g−1dgg−1ηg +Ad(g−1)dη + (−1)pg−1ηdg =

= −g∗ωG ∧Ad(g−1)(η) +Ad(g−1)dη + (−1)pAd(g−1)ηg∗ωG.

7.5. Back to connections. Recall Proposition 7.3.3, and let us reformulate it interms of the Lie algebraic data we found. Let E →M be a real vector bundle andFr(E) = P be its frame bundle. Let G = GLn(R) and g ∶= gln(R).

Proposition 7.5.1. There is a bijection between connections on E and rules thatassign a form θs ∈ A1(U,g) to a local section s of P , such that for each change-offrame function g ∶ U → G, we have

θsg = g∗ωG +Ad(g−1)θs.The curvature is given by

Ωs = dθs +1

2[θs, θs].

and the change-of-frame formula for the curvature is

Ωsg = Ad(g−1)Ωs.

Note that the above structure can be expressed entirely using the frame bundleand the Lie group/Lie algebra structure. No reference, implicit or explicit, is madeto the fact that GLn(R) is a linear Lie group. We also want to see how theconnection on E (the differential operator) can be reconstructed from these data,but we do this below and form an abstract version. We will have two versions ofthe same thing. One is an abstraction of the properties of Proposition 7.5.1. Wewill formulate it, for later use, in a slightly more general setting. The other is asingle 1-form on the total space P .

Assume that P →M is a G-principal bundle. Let (Ui, si)i∈I be a bundle atlas,Ui open and si a local section over Ui. For i, j ∈ I, we denote Uij = Ui∩Uj . There isa unique smooth function gij ∶ Uij → G such that sigij = sj . These functions satisfygijgjk = gik, the cocycle identity.

If p ∈ P is a point, the orbit map is denoted jp ∶ G → P , jp(g) ∶= xg - thisidentifies the fibres of P with G. Note that jp is G-equivariant when G carries theaction by right-multiplication.

Definition 7.5.2. A connection rule is a family θi ∈ A1(Ui,g), such that for eachi, j ∈ I, we have θj = g∗ijωG +Ad(g−1

ij )θi on Uij . A connection 1-form is an element

θ ∈ A1(P,g) such that R∗gθ = Ad(g−1)θ for all g and for all p ∈ P : j∗p = ωG.

Theorem 7.5.3. Sending the connection 1-form θ to the family (θi)i∈I , θi ∶= s∗i θdefines a bijection from the set of connection 1-forms to the set of connection rules.


Proof. We begin by figuring out what a connection 1-form on the trivial bundlelooks like. Let p ∶M ×G →M and q ∶M ×G → G be the projections and s0 ∶M →M ×G be the section x ↦ (x,1). We claim that a general connection 1-form canbe written - uniquely - as

(7.5.4) θ = q∗ωG +Ad(q−1)p∗η

for a form η ∈ A1(M,g). It is easily verified that the above formula indeed definesa connection 1-form, and the uniqueness of η is clear, since s∗0θ = η. To prove theexistence of the above formula, write η ∶= s∗0θ, put θ′ ∶= q∗ωG +Ad(q−1)p∗η and wehave to show that θ = θ′. This is a purely local problem, and we write

θ =∑i

ai(x, g)q∗ωi +∑j

bj(x, g)p∗dxj

for some scalar valued 1-forms ωi on G and dxi on M and g-valued functions.The orbit map ι ∶ G →M ×G, g ↦ (x, g) satisfies ι∗p∗dxj = 0 and q ι = idG, andso ι∗θ = ωG implies already that ∑i ai(x, g)ωi = ωG for all x ∈ G, or that

θ = q∗θG +∑j

bj(x, g)p∗ηj

The condition R∗hθ = Ad(h−1)θ for h ∈ G enforces bj(x, g) = Ad(g−1)bj(x,1). But

η = s∗0θ = ∑j bj(x,1)ηj and therefore θ has to be of the form 7.5.4.To see that θi = s∗i θ is a connection rule when θ is a connection 1-form, we have

to prove the transformation property. Since it refers to a subset of M over whichthe bundle has a cross-section, we may assume that the bundle is trivial and theconnection 1-form is given by 7.5.4. A function g ∶ M → G determines a sectionsg(x) ∶= (x, g(x)), and

s∗gθ = g∗ωG +Ad(g−1)η.Thus we can compare, using Proposition 7.4.15,

s∗ghθ = (gh)∗ωG+Ad((gh)−1)η = h∗ωG+Ad(h−1)g∗ωG+Ad(h−1)Ad(g−1)η = h∗ωG+Ad(h−1)s∗gθ

as claimed.The above computations already show that a connection 1-form is uniquely de-

termined - over U ⊂ M - by s∗θ for a section s. This proves injectivity. To showsurjectivity, we need to see that any connection rule comes from a unique connection1-form. Recall that any local section si defines a bundle isomorphism Ui×G ≅ P ∣Ui .We define a 1-form ρi ∈ A1(P ∣Ui ,g) by ρi = q∗ωG + Ad(q−1)p∗θi on Ui and trans-plant it to PUi . This is a connection form (as shown above) and s∗i ρi = θi (since thesection si corresponds to the unit section in these coordinates). We have to provethat ρi = ρj on the intersection P ∣Uij ; and this proves that the ρi glue togetherto a global form. But we have seen that a connection 1-form is determined by itspullback along one section, and thus it is enough to prove that s∗jρi = θj . In these

coordinates, the section sj is given by x↦ (x, gij(x)) and thus (g = gij)

s∗jρi = g∗ωG +Ad(g−1)θi = θjby the definition of a connection rule.

106 JOHANNES EBERT

The theorem has some useful consequences.

Corollary 7.5.5. Let f ∶ M → N be smooth, P → N and Q → M be G-principal

bundles and f ∶ Q → P be a bundle map. If θ ∈ A1(P,g) is a connection 1-form,

then f∗θ is a connection 1-form on Q.

Corollary 7.5.6. Let P → M be a G-principal bundle and φ ∶ G → H be a Liegroup homomorphism with derivative ϕ. Form the H-principal bundle Q = P ×GHand let (si) be a bundle atlas for P . Then (ti) is a bundle atlas for Q, withti(x) = [si(x),1]. Let θi ∶= s∗i θ be the connection rule. Then σi ∶= ϕ∗θi is aconnection rule for the H-principal bundle Q. Thus connections can be prolongedalong Lie group homomorphisms.

Corollary 7.5.7. Each principal bundle admits a connection 1-form.

Proof. Let λi be a partition of unity, subordinate to the open cover of a bundleatlas. Glue together local connections...

7.6. Linear connections induced by a principal connection. Now let φ ∶ G→GL(V ) be a representation, ϕ ∶ g → gl(V ) be its derivative. A local frame s for Pdefines an isomorphism

ψs ∶ Ap(U,V )→ Ap(U,P ×G V ).We define a covariant derivative on P ×G V by setting

(7.6.1) ψ−1s ∇ψsf ∶= df + ϕ(θs)f.

It is clear that the Leibniz rule holds, but that this definition is independent ofthe frame needs to be verified. We have to prove that for each function g ∶ U → G,the diagram

Ap(U,V )φ(g)⋅ //

d+ϕ(θsg)

Ap(U,V )

d+ϕ(θs)

Ap+1(U,V )φ(g)⋅ // Ap+1(U,V )

commutes and follow a form f in the left upper corner. If we map it to the lowerleft corner along the three other maps, then it becomes

φ(g)−1d(φ(g)f) + φ(g)−1ϕ(θs)φ(g)f = φ(g)−1d(φ(g))f + df + ϕ(Ad(g−1)θs)fby Lemma 7.4.9. But φ(g)−1d(φ(g)) = g∗φ∗ωGL(V ) by the computation of the

Maurer-Cartan form on GL(V ). Moreover, g∗φ∗ωGL(V ) = g∗ϕ(ωG) = ϕ(g∗ωG) byProposition 7.4.15. Therefore

φ(g)−1d(φ(g))f + df + ϕ(Ad(g−1)θs)f = df + ϕ(g∗ωG +Ad(g−1)θs)fas claimed and 7.6.1 gives indeed a well-defined covariant derivative.

Example 7.6.2. Let V → M be a Riemannian vector bundle and ∇ a metricconnection. Prove that if an orthogonal frame is chosen, then the 1-form θs takesvalues in o(n).


If P → M is a G-principal bundle, θ a connection on P and φ ∶ G → GL(V ) arepresentation, we denote the connection induced on P ×GV by ∇θ,V . The followingLemma is easy, but fundamental.

Lemma 7.6.3. Let V,W be two G-representations. Then

(1) For ω ∈ Ap(M ;P×V ) and η ∈ Aq(M ;P ×GW ), we have ∇θ,V ⊗W (ω ∧ η) =∇θ,V ω ∧ η + (−1)pω ∧∇θ,W η.

(2) If f ∶ V →W is an equivariant map, which induces a bundle homomorphismf ∶ P ×G V → P ×GW , then f is parallel, i.e. f∇θ,V ω = ∇θ,W fω.

The curvature of the induced connection ∇θ,V is called Ωθ,V . It is a 2-form withvalues in End(P ×G V ) = P ×G End(V ). By the defining formula for ∇θ,V 7.6.1, wehave in a local frame:

Ωθ,Vs = dϕ(θs) +1

2[ϕ(θs), ϕ(θs)] = ϕ(dθs +

1

2[θs, θs]).

This can be interpreted as follows: the curvature forms Ωs = dθs+ 12[θs, θs] define

a 2-form Ωθ with values in the adjoint bundle P ×G g, and since ϕ is G-equivariant,it defines a bundle homomorphism ϕ ∶ P ×G g→ P ×G End(V ). The curvature Ωθ,V

is obtained by

(7.6.4) Ωθ,V = ϕΩθ.

We can now easily prove the following fundamental result:

Theorem 7.6.5. (The Bianchi identity) The curvature Ωθ,V is parallel.

Proof. By 7.6.4 and Lemma 7.6.3, we have

∇θ,End(V )Ωθ,V = ∇θ,End(V )ϕΩθ = ϕ∇θ,gΩθ.

Therefore it is enough to prove that ∇θ,gΩθ = 0, and we do this in a local frame.The curvature is given by Ω = dθ + 1

2[θ, θ], and the connection applied to it is

dΩ + [θ,Ω]and what we have to show is thus that

d(dθ + 1

2[θ, θ]) + [θ, dθ] + 1

2[θ, [θ, θ]] = 0.

This identity holds for arbitrary g-valued 1-forms: by the graded Jacobi identity,[θ, [θ, θ]] = 0 and

1

2d[θ, θ] + [θ, dθ] = 1

2[dθ, θ] − 1

2[θ, dθ] + [θ, dθ] = 1

2[dθ, θ] + 1

2[θ, dθ].

For degree reasons, [θ, dθ] = −[dθ, θ], which concludes the proof.

7.7. The Chern-Weil construction. Now we can give the general constructionof characteristic classes for general G-principal bundles. Here are the ingredients:

(1) P →M is a G-principal bundle,(2) θ ∈ A1(P,g) a connection with curvature form Ω ∈ A2(M ;P ×G g).(3) p ∈ (g∨)⊗k is a G-invariant tensor, viewed as an equivariant map g⊗k → R

(or C). Later, we see that we can restrict to symmetric tensors.

108 JOHANNES EBERT

We know by the Bianchi identity (Theorem 7.6.5) that ∇θ,gΩ = 0. Next oneconsiders

Ω⊗k ∈ A2k(P ×G g⊗k).This is parallel, by the Bianchi identity and by the Leibniz rule 7.6.3. As p ∶

g⊗k → C is equivariant, it induces a bundle map P ×G g⊗k → P ×G C =M ×C. ByLemma 7.6.3, the 2k-form CW(θ, p) ∶= p(Ω⊗k) ∈ A2k(M) is parallel. But the trivialline bundle is induced by the trivial representation, and the connection induced onthe trivial line bundle is the exterior derivative. Hence we conclude

(7.7.1) dp(Ω⊗k) = 0.

Let us inspect the multilinear algebra involved a bit closer. The tensor algebra(g∗)⊗∗ has a product, namely two tensors p and q of degrees k and l are multipliedby the rule

p⊗ q(v1, . . . , vk+l ∶= p(v1, . . . , vk)q(vk+1, . . . , vk+l).Here we used implicitly the identification of tensors with multilinear forms. In

other words, the diagram

V ⊗k ⊗ V ⊗l

p⊗q

// V ⊗k+l

p⊗q

K⊗K // Kcommutes, where the two vertical arrows have the same name, but different

meanings.To understand the construction a little better, let us pick a local frame of P

in which the curvature is given by a form Ω ∈ A2(U,g). It follows that (p ⊗q)(Ω⊗(k+l)) = p(Ω⊗k)∧q(Ω⊗l). Before we analyze the situation closer, we summarizethe basic properties of this construction.

Theorem 7.7.2. Let P → M be a G-principal bundle, θ a connection on P andp ∈ ((g∗)⊗k)G be an invariant symmetric tensor. Then

(1) The form CW(θ, p) ∶= p(Ω⊗k) is closed.(2) If f ∶ N → M is a smooth map and f∗θ the pullback-connection, then

f∗CW(θ, p) = CW(f∗θ, p).(3) The cohomology class of CW(θ, p) is independent of θ.(4) The map p ↦ CW(θ, p) defines a homomorphism of algebras (V ∗)⊗ →Aev(M).

(5) Let φ ∶ G → H is a Lie group homomorphism and ϕ ∶ g → h its derivative.Let ϕ∗θ ∈ A1(P ×G H;h) be the prolonged connection. Let p ∈ ((h∗)⊗k)Gand ϕ∗p be the pulled back tensor. Then CW(θ,ϕ∗p) = CW(ϕ∗θ;p).

Proof. We have already proven parts (1) and (4). Part (2) is trivial (why?). Wewon’t prove part (5). Part (3) is an easy consequence of (1) and (2): Let θ0, θ1 betwo connections. Let π ∶M × [0,1]→M be the projection and t ∶M × [0,1]→ R bethe other projection. Then θ = (1−t)π∗θ0+tπ∗θ1 is a connection and j∗i θ = θi, whereji ∶ M → M × [0,1] are the inclusions. By part (1), CW(θ, p) ∈ A2k(M × [0,1])


is closed, and by part (2), CW(θi;p) = j∗i CW(θ, p). The result follows from thehomotopy invariance of the de Rham cohomology.

The tensor algebra is not very practical and we will now replace it by somethingsimpler. Let Ω ∈ A2(U,g) be the curvature. To understand the algebra, let usassume for a second that p = `1 ⊗ . . .⊗ `k is a tensor product of linear forms. Thenp(Ω⊗k) is given by

(7.7.3) `1(Ω)∧ . . .∧`k(Ω) != 1

k!∑σ∈Σk

`σ(1)(Ω)∧ . . .∧`σ(k)(Ω) = S(`1⊗ . . .⊗`k)(Ω⊗k),

(since Ω is a 2-form and 2 is even!) where S is the symmetrization operator on(g∗)k defined by

S(p)(v1, . . . , vk) =1

k!∑σ∈Σk

p(vσ(1), . . . , vσ(k)).

Let Syml(g∗) ⊂ (g∗)⊗k be the subspace of symmetric tensors, i.e. the image ofthe idempotent S. We have seen that CW(θ, p) only depends on S(p) (and on θ,but that is not the point right now). Furthermore, 7.7.3 shows that we are onlyinterested in evaluating a symmetric tensor on equal arguments.

Let Polk(V ) be the space of homogeneous polynomial functions p ∶ V → K ofdegree k. There is a map

T ∶ Symk(V ∗)→ Polk(V ); p↦ Tp; Tp(v) ∶= akp(v, . . . , v)which is an isomorphism; the inverse is given by a polarization procedure. The

composition TS is an algebra homomorphism:

(TS(p⊗q))(v) ∶= S(p⊗q)(v, . . . , v) = 1

(k + l)! ∑σ∈Σk+l

(p⊗q)(v, . . . , v) = (p⊗q)(v, . . . , v)

and

(TSp)(v)(TSq)(v) = Sp(v, . . . , v)Sq(v, . . . , v) = p(v, . . . , v)q(v, . . . , v) = (p⊗q)(v, . . . , v).The polished form of the Chern-Weil construction takes invariant polynomials

as an input.

Definition 7.7.4. Let G be a Lie group and k ∈ N. By I(G), we denote the vectorspace of degree k homogeneous polynomial functions g → C which are invariantunder the adjoint representation.

It is not recommended to try explicit computations of p(Ω) in terms of forms,local coordinates etc. We will do all relevant computations on the Lie algebra level,the only computation on a manifold was the case of the tautological line bundle onCP1, and this is already done.

Nevertheless, it is conceptually helpful to have a concrete interpretation, even ifit does not give a handy recipe for computations. As usual, Ricci calculus is thesupreme language here. So let P →M be a G-principal bundle and θ a connection.Pick a local frame, so that θ is given by a form θ ∈ A1(U ;g). The curvature is givenby dθ + [θ, θ]. The Lie bracket can be computed by means of a basis. Fix a basisX1, . . . ,Xn of g, the structure constants are defined by

110 JOHANNES EBERT

[Xi,Xj] ∶= clijXk

(Einstein convention). Write θ = θiXi; then

[θ, θ] ∶= clijθi ∧ θjXl

or

[θ, θ]l ∶= clijθi ∧ θj

(yes, if you follow the rules for the Ricci calculus, it thinks for you). The curva-ture tensor is written as Ω = ΩiXi with

Ωi = dθi + cijlθj ∧ θl.The tensor p is given, in terms of the dual base Xi of g, by

p = pi1,...,ikXi1 ⊗ . . .⊗Xik ,

the symmetry condition is expressed by the invariance of the numbers pi1,...,ikunder permutations of the indices. The G-invariance takes care of itself (!). Thefinal formula is that the 2k-form is given by

p(Ω) = 1

k!pi1,...,ikΩi1 ∧Ωik .

If you wish to know how to insert vector fields into this form, it becomes morecomplicated; likewise, picking coordinates on M given a new layer of indices.

7.8. Chern classes and the proof of the Riemann-Roch theorem. For us,the most important Lie groups are GLn(K), K = R,C, U(n), O(n) and SO(n). Wewill eventually determine the algebras of invariant polynomials in all of these cases,but we first consider complex bundles and prove the Riemann-Roch theorem. In thenext section, we give a formula for the Euler class of even-dimensional real orientedvector bundles and prove the Gauß-Bonnet-Chern theorem. A more detailed studyof the characteristic classes of real vector bundles will be undertaken in the nextsemester.

Recall that I(G) is the graded algebra of G-invariant polynomials on the Liealgebra g. A homomorphism G → H induces a map I(H) → I(G), which is com-patible with the Chern-Weil construction. We will only consider complex-valuedpolynomials. Let us begin with G = GLn(C).

Definition 7.8.1. Let ck ∈ I(GLn(C)) be defined by the formula

ck(A) ∶= ( −1

2πi)k Tr(ΛkA).

Here ΛkA is the endomorphism of ΛkCn induced by A.

Clearly, the polynomial ck has degree k. Up to factors of 2πi, ck is the k-thelementary symmmetric polynomial in the eigenvalues of A. If V →M is a vectorbundle of rank n and θ a connection, we call CW(θ, ck) =∶ ck(V ) the kth Chernclass of V . It is easy to see that c0 = 1 and that for n = 1, c1 agrees with thepreviously defined first Chern class.


Theorem 7.8.2. The homomorphism C[c1, . . . , ck] → I(GLn(C)) is an isomor-phism.

Proof. An element p ∈ I(GLn(C)) is determined by its values on the diagonalmatrices: the set of diagonalizable matrices is Zariski dense in the vector spacegln(C), and the adjoint action is given by conjugation, and so by invariance theclaim follows.

So let d(n) ⊂ gln(C) be the subspace of diagonal matrices. Any permutationof the entries can be realized by a conjugation (embed Σn as the permutationmatrices). So the restriction map I(GLn(C)) → C[x1, . . . , xn]Σn is injective. Thetheorem follows by applying the main theorem on symmetric functions.

For a vector bundle V → M , we put ci(V ) = 0 for i > rank(V ) and c(V ) ∶=∑k≥0 ck(V ) ∈H∗(M).

Theorem 7.8.3. The Chern classes have the following properties:

(1) Naturality.(2) c(V ⊕W ) = c(V )c(W ). More precisely ck(V ⊕W ) = ∑p+q=k cp(V )cq(W ).

(3) The tautological line bundle L → CP1 has Chern class c(L) = 1 − x, wherex ∈H2(CP1) is the unique element with ∫CP1 x = 1.

(4) Let Li → M , i = 0,1, be two line bundles. Then c1(L0 ⊗ L1) = c1(L0) +c1(L1).

Proof. Naturality is part of Theorem 7.7.2 and we did the computation for part(3) in example 7.2.3. For part (2), we consider the homomorphism φ ∶ GLn(C) ×GLm(C) → GLm+n(C) and let ϕ be its derivative. Let Πn ∶ GLn(C) ×GLm(C) →GLn(C) be the projection and πn be its derivative. Πm and πm are defined in asimilar fashion.

We use the functorial isomorphism Λk(V ⊕W ) ≅ ⊕p+q=k ΛpV ⊗ ΛqW and therelation Tr(A⊗B) = Tr(A)Tr(B) and compute

ϕ∗ck(A,B) = ck(A⊕B) = ( −1

2πi)k ∑

p+q=kTr(Λp(A)⊗Λq(B)) =

= ∑p+q=k

cp(A)cq(B) = ∑p+q=k

π∗ncp(A,B)π∗mcq(A,B).

In short

ϕ∗ck = ∑p+q=k

π∗ncpπ∗mcq.

Now let V = P ×GLn(C) Cn and W = Q ×GLm(C) Cm and let R ∶= ∆∗(P ×Q) bethe product GLn(C)×GLm(C)-principal bundle. Let θn be a connection on P andθm a connection on Q. Both connections together define a connection θ on R suchthat

(Πn)∗θ = θn; (Πm)∗θ = θm.The connection φ∗θ is a connection on the frame bundle of V ⊕W . Therefore

ck(V ⊕W ) = CW(φ∗θ, ck)7.7.2(5)= CW(θ,ϕ∗ck) = ∑

p+q=kCW(θ, π∗ncpπ∗mcq)

7.7.2(4)=

112 JOHANNES EBERT

= ∑p+q=k

CW(θ, π∗ncp)CW(θ, π∗mcq) = ∑p+q=k

CW((πn)∗θ, cp)CW((πm)∗θ, cq) =

= ∑p+q=k

CW(θn, cp)CW(θm, cq) = ∑p+q=k

cp(V )cq(W ).

For part (4), we proceed by a similar philosophy. Consider φ ∶ GL1(C)×GL1(C)→GL1(C), (A,B)↦ AB, and the derivative ϕ(A,B) = A+B, which gives the result.

Now we turn to the Riemann-Roch theorem.

Theorem 7.8.4. Let M be a connected Riemann surface of genus g and V → Mbe a holomorphic vector bundle. Then

ind(∂V ) = rank(V )(1 − g) + ∫Mc1(V ).

Recall that we have already done quite a bit of work: Proposition 4.6.10 statesthat there is a unique homomorphism I ∶ K0(M) → Z such that for each holomor-phic vector bundle V →M , the identity I(V ) = ind(∂V ) holds. Moreover, we haveshown by Hodge theory (Theorem 4.6.4) that

I(C) = 1 − g; I(Λ1,0T ∗M) = g − 1.

So by inspection Theorem 7.8.4 is true for the bundle C. The bundle Λ1,0T ∗Mis the dual of TM , whence

c1(Λ1,0T ∗M) = −c1(TM) = −e(TM)by Theorem 7.8.3 (4), Theorem 7.2.4. By the topological Gauß-Bonnet theorem

6.4.1, we obtain

∫Mc1(Λ1,0T ∗M) = −χ(M) = 2g − 2

and therefore

rank(Λ1,0T ∗M)(1 − g) + ∫Mc1(Λ1,0T ∗M) = g − 1 = I(Λ1,0T ∗M).

Thus the Riemann-Roch theorem holds for the two vector bundles C and Λ1,0T ∗M .The right-hand side of the Riemann-Roch formula can also be interpreted in

terms of K0: if V and W are two vector bundles on M , we get

c1(V ⊕W ) = c1(V ) + c1(W )by the product formula for Chern classes; therefore V ↦ (rank(V ), ∫M c1(V ))

defines a homomorphism

J ∶K0(M)→ Z2; V ↦ (rank(V ),∫Mc1(V )).

We have not yet proven that ∫M c1(V ) is an integer, but that is part of the nexttheorem. We will now prove:

Theorem 7.8.5. The homomorphism J takes values in Z2 and is an isomorphism,for any connected Riemann surface.


Proof of Riemann-Roch, assuming Theorem 7.8.5. Since K0(M) has rank 2, a ho-momorphism K0(M) → Z is uniquely determined by its values on these elements.We need to distinguish the two cases g ≠ 1 and g = 1. If g ≠ 1, the elements Cand Λ1,0T ∗M are linearly independent (over Q) in K0(M). If g = 1, both bun-dles have the same image in K0, and so this is not enough (in fact both bundlesare isomorphic), and we need a bundle with nonzero Chern number for which theRiemann-Roch formula is true. The bundle L(x) discussed in Theorem 4.7.4 hasChern number 1, by the Poincare-Hopf theorem. The index was computed in The-orem 4.7.4 and is 1, as desired.

Now we delve into the proof of Theorem 7.8.5

Proposition 7.8.6. Let V →M be a vector bundle of rank r > 1. Then there existsa line bundle L→M and an isomorphism L⊕Cr−1 ≅ V .

Proof. Take a section s ∶ M → V which is transverse to the zero section. Ifrank(V ) > 1, then s does not have a zero, and so V splits as V ′ ⊕ C ≅ V . Theresult follows by induction.

Corollary 7.8.7. For each vector bundle V → M , the number ∫M c1(V ) is aninteger.

Proof. The above Proposition and the sum formula reduce the statement to linebundles. But for line bundles, the first Chern class is the Euler class, which isintegral by the Poincare-Hopf theorem.

Proof of Theorem 7.8.5. Since J(C) = (1,0), it is, for the surjectivity, enough toproduce a bundle L with J(L) = (1,1). Take a map f ∶M → S2 = CP1 of degree 1.The bundle f∗H has Chern number +1.

The injectivity is more difficult (and more important for us). A general elementξ ∈ K0(M) can be written as ∑i ai[Vi], where Vi are vector bundles and ai ∈ Z.Using the relation [V ]+ [W ] = [V ⊕W ] in K0(M), we can write ξ = [V ]− [W ]. Sothe injectivity of J follows from the next claim:

(1) Let V,W → M be vector bundles with rank(V ) = rank(W ) and c1(V ) =c1(W ). Then V ≅W .

Because of Proposition 7.8.6 and the sum formula, it is enough to assume thatV and W are line bundles. Moreover, as 0 = c1(V ) − c1(W ) = c1(W ∗ ⊗ V ) =c1(Hom(W,V )), it is enough to prove that a line bundle on M with trivial Chernclass is zero. Let L be such a line bundle. By the classification theorem, thereexists a smooth map f ∶M → CPn = Gr1(Cn+1) with f∗H = L. Next, we show thatwe can assume n = 1. If n > 1, pick a regular value of f . For dimension reasons,this must be a point ` ∈ CPn ∖ f(M). Without loss of generality, we can assumethat ` = `0 = 0 × C ⊂ Cn+1. This is because the connected group U(n + 1) actstransitively on CPn. But CPn ∖ ` is diffeomorphic to Hn−1, the total space of thedual tautological line bundle on CPn−1. A diffeomorphism is given by (`, h) ↦ Γh:a linear form on some ` ⊂ Cn is sent to the graph Γh ⊂ ` × C ⊂ Cn+1. Vice versa,each line in Cn+1, with the exception of `0, is the graph of some linear form.

But Hn deformation restracts onto its zero section, namely CPn−1. So we haveproven that f ∶M → CPn is homotopic to a map M → CPn−1, if n > 1; so altogether,we may assume that there is a map

114 JOHANNES EBERT

f ∶M → CP1; f∗H ≅ L.But

∫Mc1(L) = ∫

Mf∗c1(H) = ∫

Mf∗x

with ∫CP1 x = 1 and this is the mapping degree deg(f). So we have to show that

a map f ∶M → S2 with degree 0 is nullhomotopic. This is a general fact: if Mn isa closed oriented connected manifold, then deg ∶ [M ;Sn] → Z is a bijection. Thisis a classical theorem by Hopf, the proof can be found in [51], p. 50f. The idea isvery similar to the proof that πn(Sn) ≅ Z.

7.9. Proof of the Gauß-Bonnet-Chern theorem. We now arrive at the cap-stone of the first part of this course: the Gauss-Bonnet-Chern theorem. LetM2n be a closed oriented Riemann manifold. Recall that there is the operatorD = d + d∗ ∶ Aev(M) → Aodd(M). We computed its index, and the result was, byHodge theory, the Euler characteristic of M

ind(D) = χ(M).We found two different descriptions of the Euler number. If X is a tangential

vector field on M which is transverse to the zero section, then the Poincare-Hopftheorem states that

χ(M) = ∑X(x)=0

IxX,

the sum of local indices of the vector field X. On the other hand, there isthe Euler class etop(TM) ∶= e(TM) ∈ H2n(M), and the topological Gauss-Bonnettheorem is the formula

χ(M) = ∫Metop(TM).

The Gauss-Bonnet-Chern theorem states that there is a construction of the Eulerclass in terms of the Chern-Weil construction. Above, we wrote etop for the Eulerclass that was constructed using the Thom class, which came from Poincare duality.We will now construct the class egeo, the geometric Euler class, using the Chern-Weil theorem. After the construction is done, we will prove that egeo = etop.

We have proved that the cohomology class of the Chern-Weil forms do not dependon the chosen connection. In the case of the tangent bundle of a Riemann manifold,there is a special connection, the Levi-Civita connection, and one can express theEuler form in terms of the curvature of the metric, i.e. by a geometric quantity. Inthe case of dim(M) = 2, we obtain the classical Gauß-Bonnet theorem.

Now we embark on the construction. Let V →M be an oriented Riemann vectorbundle of rank 2n, equipped with a metric connection. Let FrO(V ) → M be theoriented orthonormal frame bundle, an SO(2n)-principal bundle. We wish to findan invariant polynomial Pn on the Lie algebra so(2n) so that P (Ω) is the Eulerclass. The Lie algebra so(2n) is the space of all skew-symmetric n × n-matrices.

We have several constraints on Pn.

(1) Pn needs to have degree n.


(2) If A ∈ so(2n) and B ∈ so(2m), then Pn+m(A ⊕ B) = Pn(A)Pn(B). Thisexpresses the multiplicative property of the Euler class.

(3) A 2-dimensional oriented Riemann vector bundle is the same as a hermit-ian line bundle; this is the bundle theoretic expression of the isomorphismSO(2) ≅ U(1). There are two such isomorphisms, let us fix one, namely

φ ∶ (a −bb a

)↦ a + ib

with derivative ϕ ∶ so(2)→ u(1). Since the Euler class of a complex linebundle is the same as the first Chern class, the polynomial P1 should bethe polynomial defining the first Chern class.

We will now prove that there are unique invariant polynomials Pfn on so(2n) sat-isfying these properties. We first describe Pf1. Both Lie algebras are 1-dimensional,and the isomorphism ϕ is given by

φ ∶ ( −aa

)↦ ia.

The first Chern form is given by the linear form ia↦ −12πiia = −1

2πa, and so

Pf1((−a

a)) = −1

2πa

is the right definition. Now write Ra ∶= ( −aa

), and for a1, . . . , an ∈ R let

A(a1, . . . , an) ∶=⎛⎜⎜⎜⎝

Ra1Ra2

. . .Ran

⎞⎟⎟⎟⎠.

Multiplicativity says that we need to have

(7.9.1) Pf(A(a1, . . . , an)) =(−1)n

2nπn

n

∏i=1

ai.

Note, by the way, the identity

Pf(A)2 = 1

(2π)2ndet(A).

Now recall from Linear Algebra II that each skew-symmetric matrix is conjugateto one of the same form as A. Therefore, an invariant polynomial is uniquelydetermined by the three properties. It remains to construct the polynomial Pf.

Lemma 7.9.2. Let V be a euclidean m-dimensional vector space. Let

Φ ∶ Λ2V ∗ → so(V ); v1 ∧ v2 ↦ ⟨v1, ⟩v2 − ⟨v2, ⟩v1.

and

Ψ ∶ so(V )→ Λ2V ∗; A↦ ((v1, v2)↦ ⟨Av1, v2⟩.Then Φ and Ψ are mutually inverse equivariant isomorphisms.

116 JOHANNES EBERT

Proof. Equivariance is clear; and it is easy to calculate that ΨΦ = id. Both spaceshave the same dimension, namely 1

2m(m − 1), which completes the proof.

One calculates that

Φ(n

∑i=1

aie2i−1 ∧ e2i) = A(a1, . . . , an).

Now we define

(7.9.3) Pfn(A)vol ∶= (−1)n

n!(2π)nΦ−1(A)∧n

It is clear that Pfn is an SO(2n)-invariant polynomial on so(2n) of degree n.Note that Pfn is not invariant under the group O(2n) with the same Lie algebra;this is because in the definition, we used the orientation, more specifically thevolume form. The identity 7.9.1 follows from

Φ−1(A(a1, . . . , an))∧n = (n

∑i=1

aie2i−1 ∧ e2i)∧n = a1⋯ann!vol,

which implies the normalization and multiplicativity.

Definition 7.9.4. Let V →M be an oriented 2n-dimensional Riemann vector bun-dle, equipped with a metric connection ∇. The geometric Euler class is representedby the closed 2n-form CW(∇,Pfn) ∈ A2n(M).

Theorem 7.9.5. Let V → M be an oriented vector bundle of rank 2n. Thenegeo(V ) = etop.

We already proved Theorem 7.9.5 in the case n = 1, see Theorem 7.2.4. Theproof of Theorem 7.9.5 will be by a localization procedure. By the classification oforiented vector bundles and because the oriented Grassmann manifold is compact,it is enough to prove Theorem 7.9.5 when the base manifold M is compact.

The localization will be by means of a section. Assume that s is a section of V ,and that Z ∶= s−1(0) is compact. Let U ⊃ Z be a relatively compact neighborhood ofZ. We say that a metric connection ∇ on V is adapted if ∇ preserves the orthogonaldecomposition

V ∣M−Z = spans⊕ spans

on some open neighborhood of M ∖ U . Adapted connections exist: pick a con-nection on each of the two bundles spans and spans and take the direct sum.Then pick any connection on V ∣U and glue the connections together by means of apartition of unity.

Lemma 7.9.6. Let ∇ be an adapted connection. Then the form CW(∇,Pfn) hassupport in U .

Proof. The condition on the section s and the connection means that (outsideU) the bundle V has a reduction of the structure group to SO(2n − 1) and theconnection is induced from an SO(2n − 1)-connection. Thus it will be enough toshow that the polynomial Pfn vanishes when restricted to so(2n − 1). Recall thatPfn(A)2 = cdet(A) for a nonzero constant. But if A ∈ so(2n − 1) ⊂ so(2n), thendet(A) = 0, as desired.


Definition 7.9.7. The relative Euler class egeo(V, s) is the cohomology class (inH2nc (M)) of the form CW(∇,Pfn) of an adapted connection.

The relative Euler class is defined for each section s whose zero set is compact.It is clear that under H2n

c (M) → H2n(M), the relative Euler class maps to the(absoute) geometric Euler class. The key for the proof of Theorem 7.9.5 is

Proposition 7.9.8. Let π ∶ V → M be an oriented 2n-dimensional vector bundleand let s(v) ∶= (v, v) be the tautological section of π∗V → V . Then egeo(π∗V, s) ∈A2nc (V ) is a Thom class.

Proof of Theorem 7.9.5, assuming Proposition 7.9.8. Let t ∶ M → V be the zerosection. Then

etop(V ) = t∗τV = t∗egeo(π∗V, s) = t∗egeoπ∗V = (π t)∗egeoV = egeo(V ).

Proof. By the characterization of the Thom class, it is enough to prove that ∫Vx e(π∗V, s) =

1. But by the naturality of the Euler class, this means that it is enough to showthe relative Euler class of the trivial vector bundle R2n ×R2n → R2n, relative to theidentity section, has integral 1.

For this computation, we view R2n as Cn.The proof is completed by embedding this trivial vector bundle into a bundle

over a closed manifold, whose Euler class we can compute geometrically. Considerthe dual H → CPn of the tautological line bundle. For i = 1, . . . , n, we get a sectionsi, by taking the linear form ei on Cn+1 defining si(`) ∶= ei∣`.

Consider h ∶ Cn → CPn, (z1, . . . , zn) ↦ [1 ∶ z1 ∶ . . . ∶ zn]. A bundle chart k for Hover h(Cn) is given by

`∗ ∋ α ↦ α(1, z1, . . . , zn).Now compute

k(si(h(z1, . . . , zn))) = ksi([1 ∶ z1 ∶ . . . ∶ zn]) = k(ei∣[1∶z1∶...∶zn]) = ei(1, z1, . . . , zn) = zi.

These computations prove the following. Consider the section s = (s1, . . . , sn) ofHn → CPn. Then in the bundle chart k ⊕ . . . k, this section becomes the identitysection of Cn ×Cn → Cn. The whole point is now that the identity section extendsto a continuous section over a closed manifold. We are now ready for the finalargument. The image of egeo(Cn, id) under the map h! ∶ H2n

c (Cn) → H2n(CPn) isthe same as egeo(Hn). Therefore

∫Cnegeo(Cn, id) = ∫

CPnegeo(Hn) = ∫

CPnegeo(H)n = ∫

CPnc1(H)n = 1,

by the multiplicativity of the Euler class, Theorem 7.2.4 and Theorem 6.3.12.

7.10. Remarks. There are many sources for the Chern-Weil construction. Basi-cally, there are two approaches: one possibility is to stay in the realm of vectorbundles. You find expositions in [52], [40] and in many other places. The otherapproach is to use only principal bundles, see [44] and the superb monograph [23].One disadvantage of [23] is that he does not make a close connection to vector

118 JOHANNES EBERT

bundles. The above exposition mixes both approaches, which in my opinion isclearer.

The serious student of differential geometry needs a solid knowledge on the basicprinciples of the theory of Lie groups. Places to aquire this knowledge are the books[22], [61], [68], [26].

The proof of the Riemann-Roch theorem is my own. I do not claim that thisproof is particularly simple, but it has two advantages over the standard proofsgiven in Riemann surface texts, such as [25] or [31]. The first one is that thestandard proof only works for line bundles; whereas our proof works in a moregeneral context. This leads to the second advantage: it motivates the introductionof K-theory as the central player in index theory. In fact, the proof of the generalindex theorem on the sphere (an important special case, proven as Theorem 10.8.8)follows the same pattern.

In the treatment of the Gauß-Bonnet-Chern theorem, I followed [49], which seemsto be fairly close to Chern’s original argument.


8. Complex K-theory

8.1. Recapitulation of the definition. The part of algebraic topology that ismost relevant for the index theorem is (topological) K-theory. There exists a realand a complex version of K-theory, and the complex version is much simpler totreat. Real K-theory becomes important when one deals with real operators, butthis outside the scope of this course.

We have already defined the group K0(X) when X is a compact Hausdorff space.Recall the definition. Let Vect(X) be the abelian monoid (=semigroup) of isomor-phism classes of complex vector bundles on X. Let us remark on a slight ambiguityin the definition of a vector bundle. It is often required that the dimension of thefibres Vx, x ∈X, of a vector bundles is independent of x. When treating K-theory,we explicitly do not require this. Instead, the function X → N, x↦ dim(Vx) is onlylocally constant. If X is compact, it is true that the ranks are bounded. Of course, ifX is connected, all dim(Vx) are equal. The group K(X) is obtained from Vect(X)by the following general formal construction, the Grothendieck construction.

Lemma-Definition 8.1.1. Let M be an abelian semigroup with addition ⊕ and0. Then there is an abelian group G(M) and a semigroup homomorphism ι ∶M →G(M), such that the following universal property holds: if A is any abelian groupand f ∶ M → A a homomorphism of semigroups, then there is a unique grouphomomorphism g ∶ G(M)→ A with g ι = f .

If M is a semiring (i.e. an abelian semigroup with a distributive and associativemultiplication ⊗, then G(M) is a ring, ι is a ring homomorphism. If the above Ais a ring and f a semiring homomorphism, then g is a ring homomorphism.

Proof. There are two useful constructions of G(M). The first construction is toconsider the free abelian group ZM and divide out by the subgroup E(M) generatedby all elements (x⊕ y) − x − y, x, y ∈M . The homomorphism ι sends x to x ∈ ZM .If M is a semiring, then ZM is a ring, and E(M) a two-sided ideal.

The second construction is M ×M/ ∼. The equivalence relation ∼ is generatedby the relation (a, b) ∼ (a⊕ c, b⊕ c). M ×M/ ∼ is a monoid, and it is a group, since(x, y) + (y, x) = (x⊕ y, x⊕ y) ∼ (0,0).

We will denote, for x ∈ M , [x] ∶= ι(x) or, if there is no danger of confusion,x ∶= [x]. The monoid homomorphism M → G(M) is typically not injective.

Definition 8.1.2. Let X be a compact Hausdorff space. Then the K-theory groupof X is defined as K(X) =K0(X) = G(Vect(X)).

Lemma 8.1.3. Any element in K0(X) has a representative of the form Cn − [E].Let Ei →X, i = 0,1, be two vector bundles. Then ι(E0) = ι(E1) if and only if thereis a vector bundle E →X with E0 ⊕E ≅ E1 ⊕E.

Proof. Let [E0] − [E1] ∈ K(X). There exists an embedding E0 → Cn, and let E0

be the orthogonal complement. Then [E0] − [E1] = [E0] + [E0 ] − ([E1] + [E

0 ]) =Cn − [E1 ⊕E

0 ]. If ι(E0) = ι(E1), then there exists E such that (E0,0) + (E,E) =(E1,0) + (E,E).

Complex K-theory is a contravariant functor; pullback of vector bundles inducesa map f∗ ∶K(X)→K(Y ), for f ∶ Y →X. Homotopic maps induce the same map onK-theory; this is a consequence of homotopy invariance of vector bundles (Theorem5.1.11).

120 JOHANNES EBERT

Tensor products of vector bundles equips K(X) with the structure of a com-mutative ring with unit. It is clear that the ring structure is natural (the inducedmaps are ring homomorphisms). By the main result of Linear Algebra I, it is alsoclear that

K0(∗) ≅ Z; V ↦ dim(V ).

Definition 8.1.4. Let (X,x) be a pointed space. The reduced K-theory K0(X)is the kernel of the restriction homomorphism K0(X)→K0(x) ≅ Z.

The sequence

0→ K0(X)→K0(X)→K0(x) ≅ Z→ 0

is split exact; a splitting is given by sending n to [Cn].

8.2. Clutching constructions for vector bundles. Let π ∶ E → X be a vectorbundle and A ⊂ X be a closed subspace. We assume that there is a trivialization(π,α) ∶ E∣A → A ×Cn.

Definition 8.2.1. We consider the equivalence relation on E which is generatedby saying that e, e′ ∈ E∣A are equivalent if α(e) = α(e′). Let E/α = E/ ∼→X/A.

Lemma 8.2.2. E/α → X/A is a vector bundle. If q ∶ X → X/A denotes thequotient map, there is a natural isomorphism q∗E/α ≅ E.

Proof. It is clear that the fibres of E/α →X/A do have a vector space structure. Sothis is a family of vector space parametrized by X/A. Moreover, the quotient mapE → E/α is continuous, fibrewise an isomorphism and covers q. So what remainsto be proven is that π/ ∼∶ E/α →X/A is locally trivial. It is furthermore clear thatπ/ ∼ is locally trivial over X/A −A/A.

By Lemma 5.1.12, we can extend the trivialization α to an isomorphism (π, α) ∶E∣U → U × Cn over an open neighborhood A ⊂ U ⊂ X. Then α induces a home-omorphism and fibrewise isomorphism E∣U / ∼→ U/A ×Cn, and this is the desiredlocal trivialization of E/α over a neighborhood of A/A.

Lemma 8.2.3. The isomorphism class of E/α only depends on the homotopy classof α (by homotopy of bundle isomorphisms, we always mean homotopy throughbundle isomorphisms).

Proof. Having a homotopy of trivializations means that there is a trivialization αof (E × [0,1])∣A×[0,1] restricting to the trivializations αi at the ends of the cylinder.Thus we get a vector bundle (E × [0,1])/α →X × [0,1]/A× [0,1], by Lemma 8.2.2.The natural map F ∶ X/A × [0,1] → (X × [0,1])/(A × [0,1]) is a homotopy. It isclear that F ∗

i (E × [0,1])/α ≅ E/αi, and an application of the homotopy invarianceof vector bundles finishes the proof.

Theorem 8.2.4. Let A ⊂ X be a contractible closed subspace. Then the quotientmap q ∶ X → X/A induces a bijection Vect(X/A) → Vect(X). Hence q induces an

isomorphism K(X/A)→ K(X)

Proof. Let E → X be a vector bundle. There is a trivialization of E∣A, as A iscontractible. Two trivializations differ by composition with a bundle automorphismof A × Cn. Now as A is contractible and GLn(C) is connected, all trivializations


of E∣A are homotopic. Therefore, the map p ∶ Vect(X) → Vect(X/A); E ↦ E/α, iswell-defined by Lemma 8.2.3. By Lemma 8.2.2, the composition q∗p is the identity.

Let E → X/A be any bundle. Pick an isomorphism EA/A → Cn. This inducesa trivialization α of (q∗E)∣A and an isomorphism E ≅ (q∗E)/α. Therefore, p issurjective and hence q∗ is bijective.

Definition 8.2.5. A map f ∶X → Y of compact Hausdorff spaces is aK-equivalenceif the induced map f∗ ∶K0(Y )→K0(Y ) is an isomorphism.

Lemma 8.2.4 can be stated by saying that q ∶ X → X/A is a K-equivalence. Itis interesting to note that in general the map q is not a homotopy equivalence; andin fact, the analogous statement in singular cohomology is not true without furtherassumptions. In this sense, K-theory is better behaved than singular cohomology,but we have a price to pay: while weak equivalences induces isomorphisms in sin-gular cohomology, this is not true for K-theory. The basic reason for this differenceis that K-theory is given by entirely local data: a vector bundle is ”the same” as amap to a suitable Grassmann manifold. On the other hand, a singular cochain issomething that answers to singular simplices, and a singular simplex is by definitionspreaded over the space X. The content of the theorem on small simplices is thata cohomology class is entirely determined by its behaviour on small simplices. If Xis a metric space and ε > 0, we can for example look at the ε-small simplices, but εhas to be positive.

We can generalize the clutching construction. Let X = X0 ∪X1 be the union oftwo compact subspaces and A = X0 ∩X1, so that X = X0∐X1/A. Let Ei → Xi

be vector bundles and φ ∶ E0∣A → E1∣A be an isomorphism. We introduce anequivalence relation on E0∐E1 by E0∣A ∋ e ∼ φ(e) ∈ E1∣A. We define

E0 ∪φ E1 ∶= E0∐E1/ ∼→X.

Lemma 8.2.6. E0 ∪φ E1 is a vector bundle.

Proof. There is a natural vector space structure on the fibres, and all that is leftto be shown is local triviality. Over X ∖ A, this is clear, so let a ∈ A. There isa closed neighborhood V1 ⊂ X1 of a and a trivialization α1 ∶ E1∣V1 → V1 × Cn.By composition with φ, we get a trivialization α2 of E0∣A∩V1 . This can, again byLemma 5.1.12, be extended over an open neighborhood U0 ⊂ X0 of A ∩ V1. Nowα0 ∪ α1 is a trivialization over U0 ∪ V1, which is a neighborhood of a in X.

Lemma 8.2.7. The isomorphism class of E0 ∪φE1 only depends on the homotopyclass of φ.

Proof. This is very similar to the proof of Lemma 8.2.3.

8.3. The exact sequence. The group K0(X) is the tip of an iceberg. In fact,there is a whole sequence of functors Kn(X), n ∈ Z. It turns out that these forma generalized cohomology theory: there is a long exact sequence and excision. Theconstruction of the functors K−n(X), for n ∈ N, is by a general homotopy-theoreticconstruction, and follows from a minimal set of axioms. The construction of Kn(X)for positive n is a deep result: the Bott periodicity theorem. For the general part,we work with pointed spaces and need some homotopy-theoretic notions. Untilexplicit revocation, all spaces are assumed to be compact and Hausdorff.

122 JOHANNES EBERT

Theorem 8.3.1. Let j ∶ A → X be the inclusion of a closed subspace and q ∶ X →X/A be the quotient map. Then the sequence

K(X/A) q∗

→K(X) j∗

→K(A)is exact.

Proof. The composition A → X → X/A is constant, so the composition is null.To prove that ker(j∗) ⊂ Im(q∗), assume that E − Cn ∈ ker(j∗). This means thatthere is a bundle F → A such that E∣A ⊕ F ≅ Cn ⊕ F . By Corollary 5.1.15, thereis a complement F → A with F ⊕ F ≅ Cm. Consider the bundle E′ = E ⊕ Cm.Then E′ − Cm+n = E − Cn ∈ K(X) and E′∣A is trivial. Pick a trivialization α ofE′∣A. Then, by Lemma 8.2.2, there is an isomorphism q∗E′/α ≅ E′, and the proofis complete.

This short exact sequence is extended to a long exact sequences using a standarddevice from basic homotopy theory, the Puppe sequence.

Definition 8.3.2. Let (X,x) be a pointed space. The cone on X is the pointedspace CX =X×[0,1]/(X×1∪x×[0,1]). The wedge product X∨Y of two pointedspaces is X×y∪x×Y ⊂X×Y . The smash product is X∧Y ∶=X×Y /X∨Y . Thesuspension of X is the pointed space SX = S1∧X = [0,1]×X/[0,1]×x∪0,1×X.Note that there is a homeomorphism Sn ∧ Sm ≅ Sm+n If f ∶ Y → X is a map ofpointed spaces, the mapping cone is Cf ∶= X ∪ CY / ∼; (y,0) ∼ f(y). If X is anunpointed space, we let X+ be X with an additional basepoint, i.e. X+ =X∐x.

Note that the cone CX is contractible. Let A ⊂ X be a (closed) subspace andi ∶ A→X be the inclusion. Note that there is an inclusion j ∶X → Ci and we get a

sequence Ai→X

j→ Ci. We can extend this sequence ad infinitum:

(8.3.3) Ai→X

j→ Cik→ Cj

k→ Ck . . . .

Corollary 8.3.4. The Puppe sequence 8.3.3 induces an exact sequence

. . .→ K0(Ck)→ K0(Cj)→ K0(Ci)→ K0(X)→ K0(A)Note that this sequence cannot be extended further to the right, at least not withoutmore work.

Proof. Note that CX is contained in Ci and that Ci/CX ≅ X/A. Since CX iscontractible, we get that the quotient map ci ∶ Ci → X/A is a K-equivalence, byTheorem 8.2.4. The composition ci j ∶ X → Ci → Ci/CX = X/A is nothing elsethat the quotient map q. So, by Theorem 8.3.1, we obtain an exact sequence

K0(Ci) j∗

→ K0(X) i∗

→ K0(A).Since the Puppe sequence was obtained by repeating the same construction at

each step, this argument applies to all other terms in the sequence.

Observe that Cj = Ci∪CX. By collapsing the contractible space CX, we obtaina K-equivalence (by Theorem 8.2.4 again)

cj ∶ Cj → Cj/CX ≅ SA


(if you draw a picture, the last homeomorphism becomes obvious). Likewise(and by the same argument), there is a K-equivalence

ck ∶ Ck → Ck/CCi ≅ SX.More generally, from the term Cj on, each space in the sequence 8.3.3 contains

the cone on its precursor, and collapsing this cone yields a K-equivalence to thesuspension on its pre-precursor.

Lemma 8.3.5. The diagram

Cjl //

cj

Ck

ck

SA

Si // SX

is homotopy-commutative. Moreover, there is a natural homeomorphism SCf ≅CSf .

Proof. The space Cj is contained in S′X = X × [−1,1]/(X × ±1 ∪ x × [−1,1])(which is obviously hoemomorphic to the suspension SX). Now the map Si cj isthe composition

Cj → S′Xf0→ SX; f0(x, t) =

⎧⎪⎪⎨⎪⎪⎩

∗ t ≤ 0

(x, t) t ≥ 0

and ck l is

Cj → S′Xf1→ SX; f1(x, t) =

⎧⎪⎪⎨⎪⎪⎩

(x, t + 1) t ≤ 0

∗ t ≥ 0

and a homotopy f0 ∼ f1 is given by

(x, t)↦=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∗ −1 ≤ t ≤ −s(x, t + s) −s ≤ t ≤ 1 − s∗ t ≥ 1 − s.

The natural homeomorphism SCf ≅ CSf is easy to see (but do not try to drawa picture).

So we can replace the Puppe sequence 8.3.3 by the K-equivalent sequence

Ai→X

j→ Cid=cjk→ SA

Si→ SXSj→ S(Ci) Sd→ S2A→ . . .

Finally, SnCi→ Sn(X/A) is a K-equivalence (by Theorem 8.2.4.

Definition 8.3.6. For any pointed compact Hausdorff space X and n ≥ 0, wedefine K−n(X) ∶= K0(SnX). For an unpointed X, let K−n(X) ∶= K0(Sn(X+)) =K−n(X+).

We have proven the following important result

124 JOHANNES EBERT

Theorem 8.3.7. For each compact pointed Hausdorff pair (X,A), there is a longexact sequence

K−n(X/A)→K−n(X)→K−n(A) δ→ K−n+1(X/A)→K−n+1(X) . . .K−1(A)→ K0(X/A)→K0(X)→K0(A).Corollary 8.3.8. For each pointed space X, there is a canonical splitting K−n(X) ≅K−n(X)⊕K−n(∗).

Proof. Look at the long exact sequence of the pair (X,∗), one piece of which lookslike

⋯→ K−n(X)→K−n(X)→K−n(∗)→ ⋯The inclusion map ∗ → X is naturally split by the constant map X → ∗. This

implies that the map K−n(X)→K−n(∗) is surjective for all n. Therefore, the longexact sequence falls apart into split exact short sequences.

8.4. Relative K-theory.

Definitions. The next goal is to define relative K-groups K0(X,Y ), when Y ⊂X is

closed subspace. The long exact sequence already shows that K(X,Y ) ∶= K(X/Y )is a valid definition from a formal viewpoint. But we will often have to deal withconcrete representatives, and a more elaborate definition is necessary for this. Wewill also give an alternative definition for K−n, and will prove that all definitionsgive the same answer. The work going into this should not be considered as apunishment for deliberate multiple definitions, but as a collection of useful lemmas.Altogether, K-theory is very flexible, and the lemmas that we will show demonstratethat the essential information can be reformulated in several ways.

Definition 8.4.1. Let (X,Y ) be a compact Hausdorff pair. A K-cycle on (X,Y )is a triple (E,F,φ), with E,F → X (complex) vector bundles and φ ∶ E → F avector bundle homomorphism such that φ∣Y is an isomorphism. An isomorphismof K-cycles (Ei, Fi, φi) is a pair of isomorphisms (α ∶ E0 → E1, β ∶ F0 → F1) suchthat φ1α = βφ0. The direct sum of K-cycles (E0, F0, φ) and (E1, F1, ψ) is definedto be

(E0 ⊕E1, F1 ⊕ F1, φ⊕ ψ).We can pull back K-cycles along maps f ∶ (X,A) → (Y,B) of pairs. Namely, φ

induces a map of vector bundles f∗E → f∗F ; at x ∈ X, this is φx ∶ Ex → Fx. Aconcordance between K-cycles (E0, F0, φ0) and (E1, F1, φ1) is a K-cycle (E,F,φ)over (X × I, Y × I), together with isomorphisms (E,F,φ)∣X×i ≅ (Ei, Fi, φi). Aspecial concordance is a homotopy : if (E,F,φi), i = 0,1, are K-cycles, a homotopyφt of vector bundle homomorphisms, such that φt < −A is an isomorphism, givesrise to a concordance. A K-cycle is acyclic if φ is an isomorphism (on all of X).

Lemma 8.4.2. Concordance of K-cycles is an equivalence relation.

Proof. Let (E,F,φ) and (E′, F ′, φ′) be concordances over X × [0,1] and X × [1,2].Implicit in the definition are isomorphisms αE ∶ E∣X×1 ≅ E′∣X×1 and similarly forF . Using the clutching lemma, we get bundles E ∪αE E′ and similarly for F . Thisshows (with some obvious arguments, which only involve notational pedantism andno ideas) that concordance is transitive. Symmetry and reflexivity is trivial tocheck.


Definition 8.4.3. Let (X,A) be a compact Hausdorff pair. Let E(X,A) be themonoid of concordance classes of K-cycles on (X,A) and let D(X,A) ⊂ E(X,A)be the submonoid of those concordance classes which contain an acyclic cycle. Wedefine

K(X,Y ) =K0(X,Y ) ∶= E(X,A)/D(X,A).

Proposition 8.4.4. The monoid K0(X,Y ) is a group.

For the proof, we need a lemma that is useful in other situations as well.

Lemma 8.4.5. Let (E,F, f) and (E,F, g) be two K-cycles on (X,Y ). Assumethat

f∗g + g∗f ≥ 0 or fg∗ + gf∗ ≥ 0

holds over Y . Then these two K-cycles are homotopic.

Proof. The homotopy we look for is pt = cos(t)f + sin(t)g, t ∈ [0, π/2]. Let ε > 0 bea lower bound for f∗f and g∗g on Y , which exists by compactness. We compute

p∗t pt = cos(t)2f∗f + sin(t)2g∗g + cos(t) sin(t)(f∗g + g∗f) ≥ εThis proves that pt is injective on Y , and surjectivity follows by dimension consid-

erations. This proves the result for the second assumption. If the first assumptionis satisfied, one uses the same homotopy, but computes ptp

∗t to show that it is a

homotopy.

Proof of Proposition 8.4.4. Let (E0,E1, φ) be a K-cycle. We have to prove thatthe concordance class [E0,E1, φ] has an additive inverse. In other words, we haveto find another K-cycle so that the direct sum is concordant to an acyclic one.Pick bundle metrics on the bundles E0 and E1. We claim that [E1,E0,−φ∗] is theadditive inverse. The direct sum of both cycles is [E0 ⊕E1,E0 ⊕E1, f] with

f = ( −φ∗φ

)

It is clear that f∗ = −f and hence, for g = id, we have fg∗ + gf∗ = 0 ≥ 0, soLemma 8.4.5 applies.

It is clear that (X,Y )↦K(X,Y ) is a contravariant functor from compact pairsto groups and that it is homotopy invariant (this is built into the definition, sincewe used concordances).

Examples and the product.

Definition 8.4.6. Let M be a closed manifold, V,W → M be complex vectorbundles and D ∶ Γ(M,V )→ Γ(M,W ) be an elliptic differential operator of order k.Let π ∶ T ∗M → M be the cotangent bundle. Picking a Riemann metric on M , sothat we can talk about the unit disc bundle DM ⊂ T ∗M and the unit sphere bundleSM = ∂DM . The symbol smbk(D) is a bundle map smbk(D) ∶ π∗V → π∗W ,and the definition of ellipticity states that it is an isomorphism away from thezero section. Thus (π∗V,π∗W, smbk(D)) is a K-cycles over (DM,SM), and theresulting K-theory class σ(D) ∶= [π∗V,π∗W, smbk(D)] ∈ K0(DM,SM) is calledthe symbol class of D.

126 JOHANNES EBERT

The index theorem will express the index ind(D) in terms of the symbol classand K-theoretic operations.

Definition 8.4.7. Let α ∶ Sn−1 → GLm(C) be continuous. From α, we get a vectorbundle homomorphism Dn ×Cm → Dn ×Cm, (x, v) ↦ (x, ∣x∣α( x∣x∣)v) (as it stands,

this formula only makes sense when x ≠ 0, but it extends continuously (by 0) overthe origin. The resulting class [α] ∈ K0(Dn, Sn−1) only depends on the homotopyclass of α, and we get a map

Φ ∶ πn−1(GLm(C))→K0(Dn, Sn−1);We will show soon that Φ is a homomorphism (not entirely clear) and bijective. Aparticularly important case is α ∶ S1 → GL1(C), z ↦ z. We call [α] ∈ K0(D2, S1)the Bott class (you can guess its role!).

Lemma 8.4.8. The map Φ ∶ πn−1(GLm(C)) → K0(Dn, Sn−1) is a group homo-morphism.

Proof. On πn−1(GLm(C)) (the basepoint is the unit element), there are two groupstructure. The first one is the usual group structure on homotopy groups, andthe second is given by multiplication in GLm(C). The abstract Eckmann-Hiltonargument proves that both group structures are the same (in Topology I, we showedby a direct argument that both structures agree if n = 2. The argument generalizesto higher homotopy groups). So what we have to show is that [αβ] = [α] ⊕ [β].The point is that the two maps GLm(C) ×GLm(C)→ GL2m(C), given by

(A,B)↦ (AB1) ; (A,B)↦ (A

B)

are homotopic, the homotopy is

(A1)(cos(t) −sin(t)

sin(t) cos(t) )(1B)( cos(t) sin(t)

− sin(t) cos(t)) .

Example 8.4.9. Some elliptic operators come from a more general thing, theelliptic complexes. The passage from an elliptic complex to an elliptic operator hasa mirror in K-theory. Let (X,A) be a space pair. A Consider a chain complex E

0→ E0 α1→ E1 α2→ E2 . . .αn→ En → 0

of vector bundles over X; and require that the complex is exact over A (if n = 1,we obtain the definition of a K-cycle). Equip the bundles with hermitian metrics.Let Eev = ⊕iE

2i and Eodd = ⊕iE2i+1, and consider α + α∗ ∶ Eev → Eodd. By

an easy linear-algebraic argument (see Lemma 2.3.8), (Eev,Eodd, α + α∗) is a K-cycle. We call this K-cycle the wrapping of the complex, in symbols w(E) ∶=(Eev,Eodd, α + α∗).

It is clear that the symbol complex of an elliptic complex is such a chain complexonDM , and that the wrapping is the symbol class of the associated elliptic complex.

Let (X,A) and (Y,B) be space pairs. We define

(X,A) × (Y,B) ∶= (X × Y,A × Y ∪X ×B).The exterior tensor product of two vector bundles V →X and W → Y is


V ⊠W ∶= p∗XV ⊗ p∗YW →X × Y.Using this exterior tensor product, one can define the exterior product of two

complexes E and F of vector bundles on X and Y . Note that we have to applythe usual sign rules from homological algebra; the differential on the product isgiven by the formula ∂(a⊗ b) = (∂a)⊗ b + (−1)∣a∣a⊗ (∂b).

Definition 8.4.10. Let (E0,E1, g) and (F0, F1, f) be K-cycles on (X,A) and(Y,B), respectively. The cross product is the K-cycle

(E0,E1, g) ♯(F0, F1, f) ∶= w(E ⊠ F).More explicitly, the cross product is given by the bundles (E0 ⊗F0 ⊕E1 ⊕F1) and(E0 ⊗ F1 ⊕E1 ⊗ F0) with the map

f ♯ g ∶= (1⊗ f g∗ ⊗ 1g ⊗ 1 −1⊗ f∗) .

Lemma 8.4.11. This is a well-defined product K(X,A) ×K(Y,B) →K((X,A) ×(Y,B)).

Proof. What we have to show is that the chain complex E ⊠ F is acyclic overX × B ∪ A × Y , or that f ♯ g is an isomorphism over this subspace. One can usethe algebraic Kunneth theorem (which implies that the tensor product of two chaincomplexes over a field is exact if one of the complex is). A more direct argument:Compute

(f ♯ g)∗(f ♯ g) = (1⊗ f∗ g∗ ⊗ 1g ⊗ 1 −1⊗ f)(1⊗ f g∗ ⊗ 1

g ⊗ 1 −1⊗ f∗) = (f∗f ⊗ 1 + 1⊗ g∗g

ff∗ ⊗ 1 + 1⊗ gg∗) .

This shows that (f ♯ g)(x,y) is invertible if at least one of fx or gy is invertible.Therefore, the product is a relative K-cycle on (X,A) × (Y,B). Moreover, if onecycle is acyclic, so is the product. These facts together give the well-definedness ofthe product.

With this basic product, one can cook up other versions of the product, forexample an internal product K(X,A)×K(X,B)→K(X,A∪B), using the diagonalmap X →X ×X.

Example 8.4.12. We can generalize the Bott class. If V is a complex vector spaceof dimension n, we form the complex

0→ Λ0V → Λ1V → . . .ΛnV → 0;

of trivial vector bundles over V . The map ΛpV → Λp+1V over v ∈ V is givenby ω ↦ v ∧ ω. Now we pass to the complex conjugate bundles. It is clear that iff ∶ V → W is a C-linear map, then f ∶ V → W is C-linear as well. We denote theresulting class bV ∈K0(DV,DS). It is not hard to show that

bV ♯bW = bV ⊕W

and that bC is the Bott class.

128 JOHANNES EBERT

There is an immediate generalization of the previous example to a parametrizedsituation. Let π ∶ V → X be a complex vector bundle of rank n, over a compactHausdorff space. Consider the complex

0→ π∗Λ0V → π∗Λ1V → . . . π∗ΛnV → 0;

the resulting class in K0(DV,SV ) is called the Thom class τV . This class isrelevant for index theory, for two reasons. First, it induces the Thom isomorphism.The second, more specific reason is that when M is a complex manifold, the symbolclass of the Dolbeault complex is the Thom class of the cotangent bundle.

Some technical work and an alternative definition.

Lemma 8.4.13. Let (E0,E1, α) and (E0,E1, β) be K-cycles over (X,A). As-sume that α∣A and β∣A are homotopic (through isomorphisms, of course). Then(E0,E1, α) and (E0,E1, β) are concordant. Let (E0,E1, α) ∈ E(X,A). The follow-ing are equivalent:

(1) [E0,E1, α] = 0 ∈K0(X,A).(2) There is a vector bundle F →X, so that α∣A⊕ id∣F ∣A is homotopic (through

isomorphisms) to an isomorphism β which extends to all of X.(3) There is m ∈ N, so that α∣A ⊕ id∣Cm is homotopic (through isomorphisms)

to an isomorphism β which extends to all of X.

Proof. Consider the bundles Ei × [0,1] → X × [0,1]. Over X × 0, we take thehomomorphism α, and over X × 1, we take β. Over A × [0,1], we use a homotopybetween α∣A and β∣A. By Tietze’s extension theorem, we can extend the resultingvector bundle homomorphism over X × 0,1 ∪A × [0,1] to a homomorphism overall of X × [0,1], which yields a concordance.

Note that, by the definitions, an acyclic K-cycle is isomorphic to one of the form(F,F, idF ). Also,

By the definition of K0(X,A), (E0,E1, α) represents the zero element iff there isan acyclic K-cycle such that the direct sum is concordant to an acyclic one. By thedefinitions, an acyclic K-cycle is isomorphic to one of the form (F,F, idF ). There-fore, if [E0,E1, α] = 0, we find F → X such that (E0,E1, α⊕ idF ) is concordant toan acylic K-cycle. By the homotopy invariance of vector bundles, the concordancecan be assume to be a product, and hence α ⊕ idF is homotopic to a bundle iso-morphism. Restricting the homotopy to A× [0,1] proves that α⊕ idF is homotopicto a β which extends over X.

Vice versa, if γ ∶= α ⊕ idF is homotopic to β which extends, we build a vectorbundle homomorphism over X × 0,1 ∪ A × [0,1] using γ, β, the homotopy overA×A and the extension of β. This is an isomorphism over X×1∪A×[0,1]. Againusing Tietze, we can extend that to all of X × [0,1], which is a concordance to anacyclic cycle. These arguments show the equivalence of the first and the secondcondition. For the equivalence of the second and the third condition, one uses thatF has a complement.

Theorem 8.4.14.

(1) Let (Y, y) be a pointed space. Then

φ ∶K0(Y, y)→ K0(Y ); (E0,E1, α)↦ [E0] − [E1]is an isomorphism.


(2) Let (X,A) be a pair and q ∶X →X/A be the quotient map. Then

q∗ ∶K0(X/A,A/A)→K0(X,A)is an isomorphism.

(3) There is a natural isomorphism

K0(X/A) ≅K0(X;A).

Proof. The third part is an immediate consequence of the two other parts. To showthat φ is surjective, let [E0]−[E1] ∈ K0(Y ). By the definition of reduced K-theory,dim(E0,y) = dim(E1,y), and we pick an isomorphism αy ∶ E0,y → E1,y. Extend it,using Tietze, to a homomorphism α ∶ E0 → E1. Then

φ[E0,E1, α] = [E0] − [E1].If [E0,E1, α] ↦ 0 ∈ K0(Y ), then there is a bundle F → Y and an isomorphismβ ∶ E0 ⊕ F ≅ E1 ⊕ F . Since GLn(C) is connected, αy⊕ ∼ βy. By Lemma 8.4.13, weget that

[E0,E1, α] = [E0 ⊕ F,E1 ⊕ F,α⊕ idF ] != [E0 ⊕ F,E1 ⊕ F,β] = 0 ∈K0(Y, y).

Hence φ is injective.To show that q∗ is surjective, we begin with x = [E0,E1, α] ∈ K0(X,A). There

is a complement F → X such that E1 ⊕ F = Cn. Taking (E0,E1, α) ⊕ (F,F, idF )shows that x is represented by a cycle of the form (E,Cn, α).

Then α∣A ∶ E∣A → Cn is an isomorphism, and we can form the clutching E/α.Since α is defined over all of X, it induces a homomorphism ∣α∣ ∶ E/α → Cn ofvector bundles over X/A. By the clutching lemma, we get that

q∗(E/α;Cn; ∣α∣) = (E,Cn;α),showing that q∗ is surjective. For the injectivity of q∗, let [E,Cn, α] ∈K0(X/A,A/A)map to 0 ∈K0(X,A) (we just saw that any K-class can be represented in this way).By Lemma 8.4.13, we find m such that

q∗α⊕ id ∶ (q∗E)A ⊕Cm → Cm+n

is homotopic to β which extends to all ofX. Rename E = E⊕Cm and α = α⊕idCm

to ease notation. By the clutching lemmata 8.2.2 and 8.2.3, we get isomorphisms

E ≅ (q∗E)/α ≅ (q∗E)/(β∣A)of vector bundles over X/A. As β extends to an isomorphism over X, we get an

induced isomorphism β/(β∣A) ∶ (q∗E)/(β∣A) ≅ Cm+n of vector bundles over X/A,which shows that (q∗E)/(β∣A) and hence E are trivial. Therefore [E] − [Cm+n =K0(X/A) and by the first part of the theorem, we conclude that [E,Cn, α] = 0 ∈K0(X/A,A/A), proving that q∗ is injective.

There is another piece of basic technical work which we have to do: we give twoother descriptions of K−n(X) for n > 0. The first will be important for the proofof Bott periodicity, and the second gives an interpretation of the groups K−n(∗) ashomotopy groups πn−1(U(∞))

130 JOHANNES EBERT

Definition 8.4.15. Let X be a compact space and n > 0. An L-cycle (this is mostdefinitely not standard notation) is a pair (V, f), where V → X is a vector bundleand f is an automorphism of V × Sn−1 → V × Sn−1, such that f ∣X×∗ = id. Directsums and concordances of L-cycles are defined in the obvious way. An L-cycle isacyclic if f = id.

We let L−n(X) be defined as the monoid of all concordance classes of L-cycles,modulo the acyclic ones.

Lemma 8.4.16. L−n(X) is a group.

Proof. An inverse to (V, f) is given by (V, f−1). This is because

(cos(t) −sin(t)sin(t) cos(t) )(1

f−1)( cos(t) sin(t)− sin(t) cos(t))(f

1)

(t ∈ [0, π/2]) is a homotopy from f ⊕ f−1 to 1.

There is a map (generalizing a previous construction)

Φ ∶ L−n(X)→K−n(X);Φ(V, f) is given by the endomorphism of Dn × V of vector bundles over Dn ×X

(t, x, v)↦ (t, x, ∣t∣f( t∣t∣

)v).

Last, but not least, we consider the infinite unitary group U(∞) and homotopyclasses of maps (X × Sn−1,X × ∗)→ (U(∞),1). There is a map

Ψ ∶ [X × (Sn−1;∗)]→ L−n(X).Let g ∶ X × Sn−1 → U(∞) be given. By compactness of X, we find that g maps

into U(k), for some large k. We map g to the L-cycle (X ×Ck, f), where f is theautomorphism given by f(x,z) = g(x, z).

Theorem 8.4.17. The maps Φ and Ψ are isomorphisms.

Proof. We will show that Φ and Ψ are surjective and that Φ Ψ is injective. Φ issurjective: Let [E0,E1, α] ∈K−n(X). Let Vi ∶= Ei∣X×∗. By homotopy invariance ofvector bundles, there are isomorphisms Ei = Vi ×Dn. The bundles V0 and V1 areisomorphic, through α∣X×∗. Therefore, (E0,E1, α) is isomorphic to a cycle of theform (V ×Dn, V ×Dn, f), with f ∣X×∗ = id. Thus Φ is surjective.

Let (V, f) be an L-cycle. Pick a complement V , so that (V, f) ∼ (V ⊕V , f⊕id) =(Cn, f ′). Therefore Ψ is surjective.

Let g ∶ Sn−1 ×X → U(m) be given so that Φ(Ψ(g)) = 0. By Lemma 8.4.13, thismeans that after adding a possibly large identity matrix, g ⊕ 1 is homotopic to amap h that can be extended over all of X ×Dn. Call g ⊕ 1 just g. The inclusionSn−1 ×X → Dn ×X has the homotopy extension property, and so g extends to amap G ∶ Dn ×X → U(m) (it is easy to construct such an extension by hand, usingthe homotopy from g to h and the extension of h). There is a pointed homotopy

H ∶ [0,1] ×Dn ×X → U(m); (t, y, x)↦ G(t ∗ +(1 − t)y, x)from g to the identity.

Corollary 8.4.18. There is an isomorphism K−n(∗) ≅ πn−1(U(∞)).


Recall that there is a fibre bundle U(n) → S2n−1 with fibre U(n − 1). Thisshows that the induced homomorphism πk(U(n − 1)) → πk(U(n)) is surjective fork ≤ 2n − 2 and injective for k ≤ 2k − 3. It follows that

π2k(U(∞)) = π2k(U(k + 1)); π2k+1(U(∞)) = π2k+1(U(k + 1))

and allows the explicit computation of πi(U(∞)) for small values of i. The resultis:

Corollary 8.4.19. K0(∗) = Z; K−1(∗) = 0.

Proof. K0 is by now well-known. For K−1, compute

K−1(∗) = π0(U(∞)) = π0(U(1)) = 0

since the circle is connected. Furthermore

K−2(∗) = π1(U(1)) ≅ Z

by the knowledge of the fundamental group of the circle. We can go a little furtherby looking more closely at the fibre sequence U(1)→ U(2)→ S3 and its long exacthomotopy sequence

0 = π3(S1)→ π3(U(2))→ π3(S3)→ π2(S1) = 0→ π2(U(2))→ π2(S3) = 0

Therefore

K−3(∗) = π2(U(2)) = 0 and K−4(∗) = π3(U(2)) ≅ Z.

We will now aim at the proof of Bott periodicity, which in its most elementaryform states that

K−n(∗) =⎧⎪⎪⎨⎪⎪⎩

0 if n is odd

Z if n is even.

It is an exercise that the proof of this theorem that we will give in the sequelnever uses the (elementary) fact that π1(S1) ≅ Z. In particular, we will give anindependent (and ridiculously complicated) proof of this basic result.

8.5. K-theory with compact support and the statement of the Atiyah-Singer index theorem. There is an extension of K-theory to locally compactspaces.

Definition 8.5.1. Let X be a locally compact Hausdorff space. A compactlysupported K-cycle on X is a triple (E0,E1, α), where Ei → X are complex vectorbundles and α ∶ E0 → E1 is a vector bundle homomorphism which is an isomorphismoutside a compact set. Direct sum, isomorphism and concordances of compactlysupported K-cycles are defined in obvious analogy with Definition 8.4.1. A K-cycleis acyclic if α is an isomorphism on all of X.

132 JOHANNES EBERT

We define the compactly supported K-theory K0c (X) as the monoid of concor-

dance classes of K-cycles, modulo the acyclic ones. The proof that Kc(X) is a groupis exactly the same as the proof that K(X,A) is a group. Products in Kc(X) aredefined by the same formula as for the relative case, which yields a product

Kc(X) ×Kc(Y )→Kc(X × Y ).Some care is needed for treating functoriality and homotopy invariance. The

point is that a map f ∶ Y → X only induces a map Kc(X) → Kc(Y ) if f is proper,and that homotopy invariance only holds for proper homotopies. Since the diagonalmap X →X ×X is proper, we get an inner product on Kc(X), which turns Kc(X)into a ring without unit, unless X is compact. For compact X, there is an obviousisomorphism Kc(X) ≅ K(X). If (X,A) is a compact pair, we get a restrictionhomomorphism

K(X,A)→Kc(X −A).

Theorem 8.5.2. The homomorphism K(X,A)→Kc(X −A) is an isomorphism.

Sketch of proof. Due to excision, it is enough to consider the case (X+,∞), whereX+ is the Alexandroff (or one-point) compactification of X. We obtain a mapKc(X)→K(X+,∞) as follows: let (E0,E1, α) be a compactly supported K-cycle.Let L ⊂X be the set where α is not an isomorphism. There is a relatively compactsubset L ⊂ U ⊂ U ⊂ X. The K-cycle defines by restriction a K-cycle for the pair(U , ∂U). But by excision

K(U , ∂U) ≅ K(U/∂U) = K(X+/(X+ −U))→ K(X+,∞).It is not hard to see that this construction yields a well-defined homomorphism

Kc(X) → K(X+,∞). It is also easy to verify that this map is surjective (hint:restrict a K-cycle given on X+). It remains to be shown that the compositionKc(X)→Kc(X) is the identity.

Using this theorem, we can carry over the results for K-theory for compact spacesto this more general setting. We define

K−nc (X) ∶=Kc(X ×Rn).

The long exact sequence becomes:

Theorem 8.5.3. If Y ⊂X is a closed subspace of a locally compact space X, thereis a long exact sequence

⋯K−nc (X − Y )→K−n

c (X)→K−nc (Y )⋯

Another piece of information that we get is the following important construction:let j ∶ U ⊂X be an open subspace. There is an induced map X+ →X+/(X+ −U) ≅U+ (sic!). We get a map

j! ∶Kc(U) ≅ K(U+)→ K(X+) =Kc(X).In other words, compactly supported K-theory is a covariant functor for open

inclusion. Let us state the two main theorems of this course.


Theorem 8.5.4. (The Bott periodicity theorem) Let b = [C,C, z] ∈ Kc(R2) be theBott class. Then the map

β ∶Kc(X)→Kc(X ×R2); x↦ x ♯b

is an isomorphism, for each locally compact X.

Not one of the main results, but still important, is the following result.

Theorem 8.5.5. (The Thom isomorphism theorem) Let π ∶ V → X be a complexvector bundle of rank n, over a locally compact space X. Let

0→ π∗Λ0V → π∗Λ1V . . .→ π∗ΛnV → 0

be the chain complex, the map π∗ΛpV → π∗Λp+1V is given over v ∈ V by tykingthe exterior product with v. The Thom class tV ∈Kc(V ) is the complex conjugateof the wrap-up of this complex. Then the map

thK ∶Kc(X)→Kc(V ); x↦ π∗x ♯ tVis an isomorphism. Note that π∗x itself does not make sense (as it is not com-

pactly supported in any sense), but the prodct does make sense.

For the index formula, we do not need to know the Thom isomorphism theorem,but only the existence of the Thom class. Let M be a closed manifold. By theWhitney embedding theorem and the tubular neihborhood theorem, we can choosean embedding M ⊂ Rn; with normal bundle V → M and tubular neighborhoodU ⊂ Rn. The tangent bundle TU ⊂ TRn is an open subset and it is a tubularneighborhood of TM . The next result is easy to prove.

Proposition 8.5.6. The normal bundle of TM ⊂ TV , restricted to the zero sectionof TM , is the sum of two copies of V .

We take the first summand to be in the base-direction and the second one in the

fibre-direction. On V ⊕ V , we have the complex structure, given by J = ( −11

).

Therefore, the normal bundle of TM ⊂ TU does have a complex structure. In otherwords, there is a Thom homomorphism Kc(TM)→Kc(TU) and the map inducedby the open inclusion TU → TRn = R2n.

Definition 8.5.7. The topological index homomorphism Kc(TM) → Z is definedas the composition

top − ind ∶Kc(TM) thK→ Kc(TU)→Kc(R2n)Bott≅ Z.

Theorem 8.5.8. (The Atiyah-Singer index theorem) Let M be a closed manifoldand D an elliptic differential operator with symbol class σ(D) ∈Kc(TM). Then

ind(D) = top − ind(σ(D)).

Note that already the formulation of the index theorem refers to the Bott peri-odicity theorem. The proof of Bott’s theorem is the next major goal.

134 JOHANNES EBERT

8.6. Proof of Bott periodicity - the formal part. The proof of the Bott pe-riodicity theorem falls into two parts. The first, formal one, is due to Atiyah [4],and based on an ingenious use of the product in K-theory. Recall that the Bottperiodicity theorem states that

βX ∶Kc(X)→Kc(X ×R2);x↦ x ♯bis an isomorphism. We will now save the letter c; and denote K(X) ∶= Kc(X),

for all locally compact spaces. This should not lead to any confusion. We note thefollowing properties of βX .

(1) βX is natural in X.(2) βX is K(X)-linear, where we use left-multiplication.(3) β∗(1) = b.

We wish to produce an inverse αX ∶ K(X ×R2) → K(X) to β. As we expect βto be an isomorphism, α needs to be natural and K(X)-left-linear, and α∗(b) = 1.The main formal idea of [4] is that these requirements are enough:

Theorem 8.6.1. (Atiyah’s rotation trick) Assume that for each compact space X,there is a map αX ∶K(X ×R2)→K(X), satisfying the following axioms.

(1) αX is natural in X.(2) αX is left-K(X)-linear.(3) α∗(b) = 1.

Then we can extend the definition of α to all locally compact spaces, the Bottperiodicity theorem holds and α is the two-sided inverse to β.

For the proof, we use the multiplicative structure of K-theory heavily. Put

K−n(X) ∶=K(X ×Rn).There is a product

K−n(X) ×K−m(Y )→K(X ×Rn × Y ×Rm) ≅K−n−m(X × Y )and an interior version K−n(X) × K−m(X) → K−n−m(X), which we denote

(x,y)↦ x ⋅y. This turns K−∗(X) ∶=⊕n≥0K−n(X) into a graded ring, which is (of

course) associative and unital if (and only if) X is compact.

Proposition 8.6.2. The ring K−∗(X) is graded-commutative, i.e. if x ∈K−n(X)and y ∈K−m(X), then x ⋅ y = (−1)mny ⋅ x.

Proof. Let Tn,m ∶ X ×Rm ×Rn → X ×Rn ×Rm be the map (x, v,w) ↦ (x,w, v). Itis clear that

T +n,mx ⋅ y = y ⋅ x.Therefore, it is enough to prove that T ∗n,m is multiplication by (−1)mn. The linearmap Rm+n → Rm+n that switches the two factors has determinant (−1)mn. It istherefore enough to prove that any orthogonal self-map T of Rn induces multiplica-tion by det(T ) on K−n(X), for each locally compact space X. But T is homotopicto either the identity or the reflection in the last coordinate, depending on the signof the determinant, since SO(n) is connected. So we have to show that the reflec-tion map in the last coordinate induces −1 on K(X × Rn) = K((X × Rn−1) × R).The last equation shows that we can assume that n = 1 for this argument. Since


the inclusion K(X ×R) → K(X+ ×R) is injective, we may as well assume that Xis compact. This case is an easy exercise, which is left to the reader.

Proof of Theorem 8.6.1. In the first step, we extend α to all locally compact spaces,using the diagram

(8.6.3) 0 // K(X ×R2) //

αX

K(X+ ×R2)

αX+

// K(R2)

α∗

// 0

0 // K(X) // K(X+) // K(∗) // 0

This extended α is clearly natural with respect to proper maps. We obtainα ∶ K−n−2(X) → K−n(X), by inserting X ×Rn for X. The second step is to showthe multiplicative property of this new α. Let X and Y be two compact spaces.We claim that the diagram

K(Y )⊗K−2(X)1⊗αX //

ψ=♯

K−2(Y ×X)

αY ×X

K(Y )⊗K(X)

φ=♯ // K(Y ×X)

commutes. Since all maps are K(Y )-module homomorphisms, it is enough tocheck this on elements of the form 1⊗ x. The horizontal maps are K(Y )-linear bythe associativity of the product, and the right vertical map is K(Y )-linear by theaxioms for α. But

φ(1⊗ αX)(1⊗ x)) = φ(1⊗ αX(x)) = 1 ♯αX(x) = pr∗X(αX(x))and

αY ×X(ψ(1⊗ x)) = αY ×X(1 ♯x) = αY ×X(pr∗X×R2x)and so the commutativity of the diagram follows from the naturality of α. By

the technique used in the first step, we get the commutativity of the diagram

(8.6.4) K−m(X)⊗K−n−2(X)⋅

1⊗α

// K−m−n−2(X)

α

K−m(X)⊗K−n(X) ⋅ // K−m−n(X).

for each locally compact space X. Let now X be compact. We get, for x ∈K0(X),

α(β(x)) = α(x ♯b) = x ⋅ α(b) = xsince α(b) = 1. Also by passage to the one-point compactification, we obtain that

α β = idK(X) for all locally compact spaces, in particular X ×R2. For compact X,

we consider now x ∈K−2(X). Let us calculate

β(α(x)) 1= α(x) ♯b 2= b ♯α(x) 3= α(b ⋅ x) 4= α(x ⋅ b) 5= xα(b) = x.

136 JOHANNES EBERT

The equation 1 is the definition of β, 2 is the commutativity of the ring K−∗(X),3 that of the diagram 8.6.4, 4 is again the commutativity of K−∗(X) and 5 the left-linearity of α.

It follows that for compact X, we have β α = idK−2(X) and α β = idK(X), andthe theorem is proven for compact X. For locally compact X, use 8.6.3.

8.7. The index bundle of a Fredholm family. We turn now to the part ofthe proof of Bott’s theorem that has actually content. In the previous section, wereduced the problem to the construction of a certain map α ∶K−2(X)→K(X) forall compact X, satisfying a very short list of axioms. There are plenty of ways toconstruct such a map (in [4], other ways than the one we give are explored).

We follow a contruction that uses a deep connection between K-theory andfunctional analysis. Recall the Toeplitz index theorem. Let H ⊂ L2(S1;Cn) be theHardy space, i.e. the L2-closure of the span of the vectors zkv, k ≥ 0 and v ∈ Cn.Let P be the orthogonal projection onto H. For each map f ∶ S1 → GLn(C), wedefined the Toeplitz operator

Tf ∶= PfP + (1 − P ).

We proved that Tf is Fredholm and that ind(Tf) = −deg(det(f)), the windingnumber of the determinant function det(f) ∶ S1 → C×. Since det∗ ∶ π1(GLn(C)) →π1(C×) and deg ∶ π1(C×) → Z are isomorphisms, we find that f ↦ ind(Tf) is anisomorphism π1(GL(C))→ Z, i.e. an isomorphism K−2(∗)→ Z, and it is the inverseof the Bott map K0(∗) = Z→K−2(∗). The idea of the proof of Bott periodicity isto carry over the Toeplitz index theorem to a parametrized situation.

In short, we will use an easy index theorem (Toeplitz) to prove Bott periodicity(a topological result), and of course we will use Bott periodicity to prove a hardindex theorem (Atiyah-Singer). The connection between index theory andK-theoryactually works in both directions!!

We have to construct a map

αX ∶K−2(X)→K(X).

For that purpose, we replace both, the target and the image of α, by isomorphicgroups. Actually, these isomorphism need to be natural, and the K(X)-modulestructure needs to survive. We replace K−2(X) by the group L−2(X). There isa natural isomorphism L−2(X) → K−2(X) and it is an isomorphism of K(X)-modules. The actual content of this replacement is that we can represent each classin K−2(X) by a cycle that has a very special, simple form (loosely speaking: weneed to define α on fewer cycles), and we can reasonably expect that this is easierthan for a general cycle for K−2(X). The replacement for the target K(X) is muchmore drastic and contains one of the most substantial ideas of the whole field. Thesituation is summarized in the diagram

L−2(X) Φ //

α′

K−2(X)

α

F 0(X) K0(X).

Ξoo


All maps are natural, the horizontal maps are isomorphisms and K(X)-linear,and the diagram commutes. Instead of α, we construct α′. Now we leave thegeneral remarks behind us and jump right into this fascinating construction.

Definition 8.7.1. A Hilbert bundle over the space X is a map π ∶ E →X such thatEx ∶= π−1(x) is a separable Hilbert space and such that each point x ∈ X has anopen neighborhood U such that there is a homeomorphism h ∶ π−1(U)→ U ×H, forsome Hilbert space H which is fibrewise a linear homeomorphism of Hilbert spaces.

We do not say that a Hilbert bundle is a fibre bundle with a certain structuralgroup and fibre H. There are some problems... Let us generalize the notion of aFredholm operator to Hilbert bundles.

Definition 8.7.2. Let E0,E1 → X be Hilbert bundles. A Fredholm family is afamily (Fx)x∈X , such that Fx ∶ (E0)x → (E1)x is Fredholm operator and such thatfor all x ∈X, there exists a neighborhood x ∈ U and there exists local trivializationsh ∶ V ∣U ≅ U ×H0 and k ∶ W ∣U ≅ U ×H1, such that the map U → Fred(H0,H1),x↦ kx F h−1

x is continuous.

Remark: this is not the same as saying that F ∶ V →W , the disjoint union of allFx, is continuous. In fact, our notion is much stronger. The most obvious exampleof a Fredholm family is given by a continuous map F ∶ X → Fred(H0,H1). Thenext obvious example is given by a K-cycle (E0,E1, α): Ei is a finite-dimensionalvector bundle and in particular a Hilbert bundle, and α is a Fredholm family.

Example 8.7.3. Let V →X be a hermitian vector bundle of rank n on a compactspace. Let f be an automorphism of V × S1 → X × S1 (this is almost the same assaying that (V, f) is an L-cycle for L−2(X)). Let L2(S1, V ) → X be the Hilbertbundle ∐x∈X L

2(S1;Vx). To see that this is indeed a Hilbert bundle, we write thisas

L2(S1, V ) = Fr(V ) ×U(n) L2(S1,Cn).

Let Tf ∶ L2(S1;V )→ L2(S1;V ) be given by

Tf ∶= ∐x∈X

Tfx .

Here Tfx denotes of course the Toeplitz operator of the map fx ∶ S1 → GL(Vx).We leave it to the reader to prove that Tf is indeed a Fredholm family. Here is ahint: the problem is local in X, so one can assume that V =X ×Cn. There it boilsdown to the fact that ∥Tf∥ ≤ ∥f∥C0 , proven in 1.1.9.

There are obvious notions of direct sums, isomorphisms, pullbacks and concor-dances of Fredholm, which we won’t spell out.

Definition 8.7.4. A Fredholm family (E0,E1, F ) is acyclic if Fx is invertible forall x ∈X.

Definition 8.7.5. Let X be a compact space. By F 0(X), we denote the abelianmonoid of concordance classes of Fredholm families overX, modulo the concordanceclasses containing acyclic Fredholm families.

Proposition 8.7.6. F 0(X) is an abelian group. There is a natural structure ofa K(X)-module on F 0(X). The natural map Ξ ∶ K(X) → F 0(X) that views aK-cycle as a Fredholm family, is a homomorphism of K(X)-modules.

138 JOHANNES EBERT

Proof. Let (E0,E1, F ) be a Fredholm family. The inverse is given by (E1,E0, f∗).

Let

gt ∶= (F t−t F ∗) ∶ E0 ⊕E1 → E1 ⊕E0.

For t = 0, one gets the direct sum. For t ≠ 0, gt is invertible by the followingargument:

g∗t gt = (FF∗ + t2

F ∗F + t2)

The entries in the diagonal are selfadjoint and positive definite, hence injective.Moreover, FF ∗ + t2 ≥ t2, and so FF ∗ + t2 is bounded from below and hence hasclosed image. The orthoginal complement of the image of a self-adjoint operator isthe kernel, and hence FF ∗+t2 is bijective. The same argument applies to F ∗F +t2,which shows that gt is injective and g∗t surjective. Reversing the roles of gt and itsadjoint proves that gt is invertible.

The point of the module structure is that one can form the tensor product E⊗Vof a Hilbet bundle with a finite dimensional vector bundle without problem (thereis a notion of the tensor product of two Hilbert bundles, but that is more tricky).The verification of the module property and linearity is trivial.

The main theorem about these things is

Theorem 8.7.7. The natural map Ξ ∶K(X)→ F 0(X) is an isomorphism.

From this theorem, the proof of Bott periodicity will be very short, but we need adescription of the inverse map to Ξ. The inverse of Ξ is a map ind ∶ F 0(X)→K(X),which takes the family index of a Fredholm family over a compact space X. Thedefinition is based on two lemmata.

Lemma 8.7.8. Let Hi → X, i = 0,1, be Hilbert bundles and F ∶ H0 → H1 be aFredholm family which is pointwise surjective. Then ∪x∈X ker(Fx) ⊂ H0, with thesubspace topology, is a (finite-dimensional) vector bundle.

Proof. (this is a straightforward generalization of Lemma 5.1.6) This is a localproblem, which is why we can assume that both Hi are trivial and F is given bya map X → Fred(H0,H1). Let o ∈ X. Let P ∶ H0 → ker(Fo) be the orthogonalprojection. The map G ∶H0 →H1⊕ker(Fo), v ↦ (Fxv,Pv) is an isomorphism at o,and therefore also in a small neighborhood. Since taking inverses is continuous (evenfor Hilbert spaces), the map G−1 is also an isomorphism. But Gx(kerFx) = ker(Fo),and this is the desired local trivialization.

Lemma 8.7.9. Let F ∶H0 →H1 be a Fredholm family on a compact base space X.Then there exists a finite-dimensional bundle V →X and a bundle map g ∶ V →H1

such that F +g ∶H0⊕V →H1 is surjective. We call such a bundle a taming bundle.

Proof. Let o ∈ X and Vo = Im(Fo). The map (H0)o ⊕ Vo → (H1)o; (v,w) ↦Fo(v) + w is surjective. Extend Vo → H1 somehow to a bundle map go ∶ Vo → H1,which is surjective on an open neighborhood Uo of o (since surjectivity of boundedoperators is an open condition). By compactness of X, there are finitely manyx1, . . . , xr ∈X such that ⋃iUxi =X. Consider


V ∶=⊕i

Vxig=∑gi→ H1

and F + g is surjective.

Definition 8.7.10. Let F ∶ H0 → H1 be a Fredholm family and let g ∶ V → H1

be a map from a finite-dimensional bundle to H1 such that F + g ∶ H0 ⊕ V → H1,(v,w)↦ Fv + gw is surjective. We define the index bundle of F as

ind(F ) ∶= [ker(F + g)] − [V ] ∈K0(X).

Proof that this is well-defined. By Lemma 8.7.9, the data (V, g) exists. To showthat the index bundle does not depend on the choice of (V, g), let gi ∶ Vi → H1 betwo such choices. Consider for s, t ∈ [0,1]:

F (s, t) ∶H0 ⊕ V0 ⊕ V1 →H1; (v, x, y)↦ Fv + sg0(x) + tg1(y).If t or s is not zero, the map F (s, t) is surjective. Thus t ↦ F (1 − t, t) is a

homotopy of surjective Fredholm families. Therefore the kernel bundles of F (0,1)and F (1,0) are concordant, hence, by homotopy invariance of vector bundles, iso-morphic. On the other hand, ker(F (0,1)) = V0 ⊕ ker(F + g1) and ker(F (1,0)) =V1 ⊕ ker(F + g0) and

V0 ⊕ ker(F + g1) ≅ V1 ⊕ ker(F + g0)follows, or

[ker(F + g0)] − [V0] = [ker(F + g1)] − [V1] ∈K0(X).

Remark 8.7.11. For each x ∈ X, the composition F (X) → K(X) → K(x) = Ztakes [H0,H1, F ] to the ordinary Fredholm index ind(Fx).

The proof yields some corollaries for free.

Corollary 8.7.12.

(1) Let H,H ′ → X × [0,1] be Hilbert bundles and F ∶ H → H ′ be a Fredholmfamily, with restrictions Fi to X × i. Then ind(F0) = ind(F1) ∈K0(X).

(2) Let f ∶ Y →X be continuous. Then f∗ ind(F ) = ind(f∗(F )) ∈K0(Y ).

Proof of Theorem 8.7.7. We have constructed two homomorphisms Ξ ∶ K(X) →F 0(X) and ind ∶ F 0(X) → K(X). We will prove that ind Ξ = id and that ind isinjective.

The group K(X) is generated by K-cycles of the form (W,0,0). Consider sucha thing as a Fredholm family. For the vector bundle W → X as in Lemma 8.7.9,we can take the zero bundle. As the index bundle does not depend on the choiceof the additional bundle by 8.7.10, we find that ind Ξ[V,0,0] = V .

To show that ind is injective, take a Fredholm family (H0,H1, F ) with ind(H0,H1, F ) =0. Let g ∶ V →H1 be a taming bundle. That the index is zero means that theK-class[ker(F + g)] − [V ] is zero, or, in other words, that there exists a finite dimensionalbundle W →X such that

ker(F + g)⊕W ≅ V ⊕W.

140 JOHANNES EBERT

We replace the taming bundle g by (g,0) ∶ V ⊕W → H1. But ker(F + (g,0)) =ker(F + g)⊕W ≅ V ⊕W . The arguments so far show that we can assume that thetaming h ∶ U → H1 is chosen in such a way that ker(F + h) ≅ U . Now we assumethis (and switch notation back to g ∶ V →H1) and put

R ∶H0 ⊕ Vpr→ ker(F + g) ≅ V

using an isomorphism ker(F + g) ≅ V . For each x ∈ X, the operator Rx iscompact, because it has finite-dimensional image. We write R = (R0,R1), withR0 ∶H0 → V and R1 ∶ V → V . In F 0(X), we have the equality

[H0,H1, F ] = [H0 ⊕ V,H1 ⊕ V,F ⊕ 1].The difference of the operators

F0 = (F1) and F1 = ( F g

R0 R1)

is (pointwise in X) compact. Therefore (1−t)F0+tF1 is a homotopy of Fredholmfamilies (this uses Corollary 1.5.2 (1)), and we get that

[H0 ⊕ V,H1 ⊕ V,F ⊕ 1] = [H0 ⊕ V,H1 ⊕ V,F1] ∈ F 0(X).Finally, the operator F1 is bijective: if (h, v) ↦ 0, then F (h) + g(v) = 0 and

R(h, v) = 0. The latter means that (h, v) ∈ ker(F + g) and the former that (h, v) ∈ker(F + g), so altogether F1 is injective. By remark 8.7.11, the operator (F1)x hasindex zero for all x ∈X and is injective, and so surjective. Therefore F1 is invertibleand [H0 ⊕ V,H1 ⊕ V,F1] = 0 ∈ F 0(X).

Corollary 8.7.13. The map ind ∶ F (X) → K(X) is a homomorphism of K(X)-module homomorphisms.

Conclusion of the proof of Bott periodicity. We construct a map α′ ∶ L−2(X) →F 0(X) as follows. Let (V, f) be an L-cycle. Let

α′([V, f]) ∶= [L2(S1, V ), L2(S1, V ), Tf ].We need to show that α is well-defined. Besides saying that concordant L-cycles

give concordant Fredholm families (which follows from the continuity f ↦ Tf ), thisrequires showing that acyclic L-cycles give acyclic Fredholm families. Acyclicity of(V, f) means that f = id, and this means that Tf = id, in particular, it is invertible.Naturality of α is clear. One has to check that α is K(X)-linear. The point is thatif W → X is another finite-dimensional bundle, then Tf⊗W = Tf ⊗ 1W . Finally, weneed to check that α′ maps the Bott class to 1. Let b′ = [C, z ↦ z] ∈ L−2(∗). Underthe natural map L−2 →K−2, this maps to the Bott class. In view of remark 8.7.11,all that remains to be done is to prove that the (ordinary) Fredholm index of Tz isequal to 1 (this shows that α′(b′) = ψ(1) ∈ F 0(∗) or that

α(b) ∶= ind(α′(Φ(b′))) = 1

as desired. But we computed the Fredholm index of Tz, and the result is indeed+1, compare the computation before the Toeplitz index theorem 1.1.11.


8.8. Historical remarks and references. The basic material on K-theory istaken from Atiyah’s classical monograph [3], which is still unmatched in its elegance.The literature on Bott periodicity is a bit more twisted. Originally, Bott provedthe periodicity theorem when he was interested in homotopy groups of the stableunitary (and orthogonal) group. His proof [16] was differential-geometric, and usedMorse Theory on loop spaces. There is no mentioning of K-theory in his paper.A good reference for Botts proof is Milnors book [50]. Topological K-theory wasintroduced by Atiyah and Hirzebruch [7]. Later on, Atiyah and Bott worked (inconnection with the index theorem) on an elementary proof [10]. This is the proofthat is reproduced in Atiyahs classical monograph [3] and most later expositions, as[40] and [33]. The formal framework for the proof of Bott periodicity was introducedby Atiyah as well [4]. The use of the Toeplitz operators is essentially taken fromthat paper and from Atiyah’s survey [5], but the details are not there. The proof in[10] was generalized by Wood [71] and it is definitely recommended to study Woodsproof. A similar proof was given by Karoubi [43]. However, it is unclear to me howone can use Karoubi’s approach to K-theory in connection with the index theoremin a fruitful way.

If you are interested in the deeper aspects of index theory, you must know K-theory of C∗-algebras. It contains topological K-theory as a special case, and thealgebraically inclined reader might prefer to study C∗-algebras from the beginning.Sources to start are [69], [36].

The relation betweenK-theory and Fredholm operators can be sharpened consid-erably. A theorem by Atiyah and Janich says that the space of Fredholm operatorsis a classifying space for K-theory. The main ingredient, beyond what we use in thischapter, is Kuiper’s theorem [45] that the unitary group of an infinite-dimensionalHilbert space is contractible. For the Atiyah-Janich theorem, see the appendix to[3], [41] and [15].

The formulation of the index theorem is the one that is given in the originalpaper by Atiyah and Singer [8].

142 JOHANNES EBERT

9. Advanced theory of characteristic classes

9.1. Recapitulation of the Chern-Weil construction. The basic constructionof characteristic classes, the Chern-Weil construction, was carried out in the lastterm. Let us recall some of the main points. Let G be a Lie group and g its Liealgebra. Let P →M be a smooth G-principal bundle. We proved that there exists aconnection on P , which by definition is a form θ ∈ A1(P ;g) with certain properties.The curvature of θ is the form Ω = dθ + 1

2[θ, θ] ∈ A2(P ;g). We interpreted Ω as a

2-form on M with values in the bundle P ×G g, where G acts on g by the adjointrepresentation. Let Ik(G) = Symk(g∗)G be the vector space of polynomials F ∶ g→R of degree k which are invariant under the adjoint representation, F (Ad(g)X) =F (X), for all X ∈ g and g ∈ G, and I(G) =⊕k≥0 Ik(G). By the invariance condition,we can form

F (Ω) ∈ A2k(M).The central fact in the whole story is that the form F (Ω) is closed. Therefore,

there is the cohomology class

CW(θ,F ) ∶= [F (Ω)] ∈H2kdR(M).

It followed quite easily from the fact that F (Ω) is closed that the cohomologyclass CW(θ,F ) only depended on P , not on the choice of the connection. Let uswrite

CW(P,F ) ∶= CW(θ,F )for a connection θ. This construction had some important naturality properties,

which we summarize. Let f ∶ N →M be a smooth map, which gives a bundle mapf∗P → P over f , also denoted f . The form f∗θ ∈ A1(f∗P,g) is a connection onf∗P , and

CW(f∗θ,F ) = f∗CW(θ,F )holds. Moreover the Chern-Weil construction is multiplicative:

CW(P,EF ) = CW(P,E)CW(P,F )for E,F ∈ I(G). Let φ ∶ G → H be a Lie group homomorphism, with derivative

dφ ∶ g→ h. We get a map φ∗ ∶ I(H)→ I(G), F ↦ f dφ. If P →M is a G-principalbundle, then P ×GH →M is a H-principal bundle, denoted φ∗P . We showed thatfor all F ∈ I(H):

CW(φ∗P,F ) = CW(P,φ∗F ).Using these properties, another multiplicativitiy follows easily. Let G and H be

two Lie groups, P →M a G-bundle and Q→ N be an H-bundle. We can form theG ×H-bundle P ×Q →M ×N . Let πG ∶ G ×H → G be the projection onto G andπH defined accordingly. We obtain a map

× ∶ Ik(G)⊗ Il(H)→ Ik+l(G ×H);F ×E ∶= (π∗GF )(π∗HE).In plain language: if X ∈ g and Y ∈ h, then (X,Y ) ∈ g × h and (F ×E)(X,Y ) =

F (X)E(Y ). We compute


CW(P ×Q;F ×E) = CW(P ×Q, (π∗GF )(π∗HE)) = CW(P ×Q,π∗GF )CW(P ×Q,π∗HE) =CW((πG)∗(P ×Q), F )CW((πH)∗(P ×Q),E) = CW(P,F )CW(Q,E).

(9.1.1)

9.2. The universal perspective. We did not prove it, but mentioned the classi-fication of principal bundles. For a topological space X, we let PrinG(X) be the setisomorphism classes of G-principal bundles on X. There is a ”universal” principalbundle EG→ BG such that for each paracompact Hausdorff space X, the map

[X;BG]→ PrinG(X); f ↦ f∗EG

is a bijection. We denote the unique (up to homotopy) map which correspondsto P by fP , so that f∗PEG ≅ P . The bundle EG → BG is unique up to homotopyequivalence, and it is characterisized by the property that EG is contractible.

Definition 9.2.1. Let G be a Lie group. A characteristic class for G-bundlesof degree k with coefficients in the ring R is a natural transformation of functorsPrinG( )→Hk( ;R) from the category of paracompact spaces to sets.

A well-known triviality from category theory (the Yoneda lemma) shows thatthe set of characteristic classes is in bijection with Hk(BG,R). More precisely,to a characteristic class c, we assign c(EG) ∈ Hk(BG;R) and to x ∈ Hk(BG;R),we assign the characteristic class that takes a bundle P → X to f∗Px ∈ Hk(X;R).We wish to say that the Chern-Weil construction defines a ring homomorphismI(G)→H∗(BG;R) in this way, but there is an obtacle to overcome: the Chern-Weilconstruction only works for manifolds, and BG is most definitely not a manifold(at least not of finite dimension). But first, we note some consequences of theclassification theorem.

We will work with compact Lie groups all the time. For such G, it is no problemto get a space BG which is paracompact. Let φ ∶ G→H be a group homomorphism.Then EG ×GH → BG is an H-principal bundle, and so it has a classifying map

Bφ ∶ BG→ BH

and it is not hard to see, using the universal property, that G ↦ BG is indeed afunctor from groups to the homotopy category of paracompact spaces. If H ⊂ G isa closed subgroup of a compact Lie group, then EG → EG/H = EG ×G G/H is auniversal H-principal bundle. Clearly B1 ≃ ∗, and it is an exercise to show that

B(G ×H) ≃ BG ×BHholds. For later use, we need a lemma.

Lemma 9.2.2. Let G be a Lie group and g ∈ G. Let cg ∶ G → G be automorphismh↦ ghg−1. Then Bcg ∶ BG→ BG is homotopic to the identity.

Proof. Let P → X be a G-principal bundle. The G-bundle P g ∶= P ×G,cg G → X is

given as G/(P ×G), where G acts from the left as h(p, k) ∶= (ph−1, ghg−1k). Theright G-action that defines the principal structure is (p, k)h = (p, kh). The mapP ×G→ P , (p, h)↦ pg−1k descends to a bundle map P g → P covering the identity.Thus P g is isomorphic to P .

144 JOHANNES EBERT

The map Bcg ∶ BG→ BG is, by definition, a classifying map for the bundle EGg,and since EGg ≅ EG, Bcg also classifies the bundle EG. Thus, by the homotopyuniqueness of classifying maps, Bcg is homotopic to the identity.

We now explain how to construct a homomorphism I(G) → H∗(BG ∶ R) fromChern-Weil theory. LetA be the set of natural transformations PrinG( )→H∗( ;R),of functors defined on the category of all (paracompact) spaces. Let PrinG,C∞( )be the functor that assigns to a smooth compact manifold the set of isomorphismclasses of smooth G-principal bundles. Let B be the set of natural transformationsfrom the functor PrinG,C∞( ) to H∗( ;R), defined on the category of smooth com-pact manifolds (compactness will be convenient later, but not strictly necessary).By restriction, we get a map

A→ B.(the only content is that we can look at characteristic classes only for smooth

bundles).

Theorem 9.2.3. The map A → B is a bijection (in fact, a ring isomorphism).Therefore, we get I(G)→H∗(BG;R).

In plain words, a characteristic class defined for smooth G-bundles on compactmanifolds uniquely extends to a characteristic class defined on all bundles overparacompact spaces.

Proof. Recall that we assumed G to be compact. By the Peter-Weyl theorem,there is an injective homomorphism G→ U(k), for some k (for the classical groups,one does not have to invoke the Peter-Weyl theorem at all). So we can assumethat G ⊂ U(k). Recall the Stiefel manifold Stk(Cn) and the Grassmann manifoldGrk(Cn) = Stk(Cn)/U(n). The quotient map Stk(Cn) → Grk(Cn) is a U(n)-principal bundle., in fact it is the frame bundle of the tautological vector bundle.The Stiefel manifold is the homogeneous space

Stk(Cn) = U(n)/U(n − k).Recall that πi(U(n−k))→ πi(U(n)) is surjective for i ≤ 2(n−k)+1 and injective

for i ≤ 2(n − k). Therefore πi(Stk(U(n))) = 0 for i ≤ 2(n − k). Now we defineEnG = Stk(Cn) and BnG = EnG/G. The map EnG → BnG is a smooth U(n)-principal bundle over a compact manifold. Let fn ∶ BnG → BG be a classifyingmap for this bundle. Now look at the diagram

Gid //

G

EnG

// EG

BnG

fn // BG.

This is a pullback and the columns are fibrations. By the long exact sequence andthe 5-lemma, we conclude that (fn)∗ ∶ πi(BnG) → BG is bijective for i ≤ 2(n − k).By the Hurewicz theorem and the universal coefficient theorem, we get that


f∗n ∶Hi(BG;R) ≅Hi(BnG;R) for i ≤ 2(n − k).There are inclusion jmn ∶ BnG → BmG, and by the uniqueness of classifying

maps, fm jmn = fn. If P → M is a smooth G-principal bundle over a closedmanifold, consider the U(k)-principal bundle P ×G U(k). There is a G-equivariantmap P → P ×G U(k), given by p ↦ [p,1]. By the classification of vector bundles,there is a bundle map P → Stk(Cn), for some n >> dim(M). Therefore, there isa G-map P → EnG. The last argument proves that each G-bundle on a closedmanifold is induced by a map M → BnG, and so the classifying map factors as

M → BnGfn→ G.

The above arguments form the heart of the theorem and the proof is finishedby formal nonsense. Injectivity of the map A→ B means that if two characteristicclasses x1, x2 of degree i defined for all spaces agree on closed manifolds, thenthey agree for all bundles. But f∗nx1(EG) = x1(EnG) = x2(EnG) = f∗nx2(EG).Chosing n large enough (i.e. i ≤ 2(n − k)) shows that x1(EG) = x2(EG); i.e.x1 = x2. For surjectivity, let y be a characteristic class of degree i defined on closedmanifolds. Pick n with i ≤ 2(n − k). Then f∗n ∶ Hi(BG;R) → Hi(BnG;R) is anisomorphism, so that there is a unique x ∈ Hi(BG;R) with y(BnG) = f∗n(x). Foreach m ≥ n, (jmn )∗f∗mx = f∗nx. Because (jmn )∗ is an isomorphism on Hi, we findthat f∗mx = y(EmG). So the characteristic class defined by x agrees with y on allEmG. If P →M is an arbitrary bundle on a closed manifold, use a classfying mapM → BmG to find that x agrees with y on all manifolds.

9.3. Chern, Euler and Pontrjagin classes: the invariants of the classicalgroups.

The circle. The circle is the Lie group T = S1 ⊂ C; also isomorphic to R/Z andSO(2). Its Lie algebra is t = iR. We define a basic element in t∗ = I1(T), by

x(z) = −1

2πiz ∈ R.

Clearly I(T) = R[x]. Let P →M be a T-principal bundle and L ∶= P ×T C →Mbe the associated line bundle (we take the action by the identity). The first Chernclass of L was by definition

c1(L) = CW(P,x) ∈H2(M).Let H → CPn be the dual tautological line bundle. The one fundamental com-

putation from which ultimately everything derives is:

(9.3.1) H∗(CPn) = R[c1(L)]/(c1(L)n+1); ∫CPn

c1(H)n+1 = 1.

More generally, we consider the n-fold cartesian product T(n) of the circle withitself. We define xj(z1, . . . , zn) ∶= x(zj), xj ∈ I1(T(n)). Of course I(T(n)) =R[x1, . . . , xn]. An important property of the first Chern class was its behaviourunder tensor products of line bundles, namely c1(L0 ⊗ L1) = c1(L0) + c1(L1). Inthe abstract framework of Chern-Weil theory, this can be seen as follows. Letµ ∶ T(2) → T be the homomorphism µ(z1, z2) = z1z2. Then dµ(t1, t2) = t1 + t2 andtherefore

146 JOHANNES EBERT

µ∗x = x1 + x2.

It is an exercise with the formal properties of the Chern-Weil construction toderive the tensor product formula.

Theorem 9.3.2. The universal Chern-Weil homomorphism I(T(n))→H∗(BT(n);R)is an isomorphism.

Proof. We use the notation and results from the proof of Theorem 9.2.3. Firstconsider the case n = 1. Using the embedding T ⊂ U(1), we obtain BmT = CPm−1.We look at the composition

I(T)→H∗(BT;R)→H∗(BmT;R).The composition is an isomorphism in degrees ∗ ≤ 2m − 1, by the fundamental

result 9.3.1. The second map is an isomorphism in roughly the same range ofdegrees, by the arguments in the proof of Theorem 9.2.3. Therefore, the first mapis an isomorphism in this range, and as we can pick m as large as we want, anisomorphism in all degrees.

The case n > 1 is derived from the n = 1 case and the Kunneth theorem. Considerthe commutative diagram

I(T)⊗n //

H∗(BT)⊗n //

H∗(CPm−1)⊗n

I(T(n)) // H∗(BT(n)) // H∗((CPm−1)n).

The diagram commutes by the product formula for the Chern-Weil classes 9.1.1.The right vertical map is an isomorphism by the Kunneth theorem. The horizontalmaps on the left are both isomorphisms in a range of degrees, and so the middlevertical map is an isomorphism in these degrees. The left vertical map is (obvi-ously) an isomorphism, and in the first part of the proof we showed that the upperleft horizontal map is an isomorphism. Therefore, I(T(n)) → H∗(BT(n)) is anisomorphism in a range increasing with m.

The unitary group. In the first part of this course, we already learnt about theChern classes. The Chern class for n-dimensional complex vector bundles is givenby the invariant polynomial

ck(A) ∶= (−1)k

(2πi)kTr(ΛkA) ∈ Ik(GLn(C)).

There are more convenient forms of this definition. The elementary symmetricpolynomials σk ∈ Z[x1, . . . , xn] are defined by the relation (in Z[t, x1, . . . , xn])

n

∏i=1

(t + xi) =n

∑k=0

tn−kσk(x1, . . . , xn)

or

n

∏i=1

(1 + txi) =n

∑k=0

tkσk(x1, . . . , xn) =∞∑k=0

tkσk(x1, . . . , xn).


The last equation simply expresses the convention that σk(x1, . . . , xn) = 0 ifk > n. Let λ1, . . . , λn be the eigenvalues of A. Using the easily verified formula

n

∏j=1

(1 + tλj) = det(1 + 1A) = ∑k≥0

tk Tr(ΛkA),

we arrive at the following expression

ck(A) = σk(−1

2πiλ1, . . . ,

−1

2πiλn),

where the λj are the eigenvalues of A (as usual, counted with multiplicity).We now restrict our attention to the maximal compact subgroup U(n) ⊂ GLn(C).Inside U(n), we have the group (S1)n ≅ T(n) ⊂ U(n) of diagonal matrices. The Liealgebra of U(n) is u(n), the Lie algebra of skew-hermitian matrices, and the Liealgebra t(n) of T(n) is the (abelian) Lie algebra of diagonal matrices with purelyimaginary entries. Define linear forms xj ∈ t(n)∗ by

xj(diag(λ1, . . . , λn)) ∶=−1

2πiλj

(note that this is real-valued!), so that we obtain the important relations

ck ∣t(n) = σk(x1, . . . , xn).and

c =∞∑t=0

tkck =n

∏j=1

(1 + txj).

In particular, ck ∣u(n) is real-valued. Using the description using eigenvalues, wecan easily derive

(9.3.3) c((AB)) = c(A)c(B).

Using the Chern-Weil construction, we obtain for each hermitian vector bundleV →M of rank n (alias U(n)-principal bundle), the Chern classes

ck(V ) ∶= CW(Fr(V ), ck) ∈H2k(M ;R)and the total Chern class:

c(V ) = 1 + c1(V ) + c2(V ) + . . . + cn(V ) ∈H∗(M ;C).Here are the important properties, all proven in the last term.

Theorem 9.3.4. The Chern classes have the following properties:

(1) c(V ⊕W ) = c(V )c(W ).(2) For two line bundles V0, V1, we have c1(V0 ⊗ V1) = c1(V0) + c1(V1).(3) ck(V ) = (−1)kck(V ).(4) Let H → CPn be the dual tautological line bundle. Then, for k ≤ n,

H2k(CPn) is 1-dimensional with basis c1(H)k. Furthermore ∫CPn c1(H)n =1.

148 JOHANNES EBERT

The first part is a consequence of 9.3.3, the second was proven last term and thethird follows from the obvious equation

(9.3.5) ck(A) = (−1)kck(A).The fourth was a fundamental computation and has real content. Another fun-

damental fact is the first integrality theorem, proven in the exercises.

Theorem 9.3.6. Let F ∈ Z[x1, . . . , xn] and let V →M be a complex vector bundleover a closed oriented manifold. Then

∫MF (c1(V ), . . . , cn(V )) ∈ Z.

The ultimate source for this integrality is that the integral counts a certain in-tersection number. The Bott perioditicity is another profound source of integralitytheorems, and the integrality comes from the fact that the integral is an index.

Next, we use a bit of classical invariant theory and determine the structureof the algebra I(U(n)). It is clear that I(T(n)) = R[x1, . . . , xn]. Note that thesymmetric group Σn acts on T(n) by permutation of the entries. Hence Σn actsalso on I(T(n)), by permuting the generators xj . The restriction of the Chernpolynomial is symmetric, i.e. Σn-invariant.

Theorem 9.3.7. The maps R[c1, . . . , cn]→ I(U(n))→ I(T(n))Σn = R[x1, . . . , xn]Σnare isomorphisms.

Proof. The first thing we have to prove is that the restriction of an element inI(U(n)) to t(n) is indeed invariant under the action of Σn. To this end, usethe action of Σn on Cn by permuting coordinates. This given a homomorphismρ ∶ Σn → U(n). Through the adjoint representation, Σn acts on u(n), and it leavesthe subspace t(n) invariant, more precisely

Ad(ρ(s−1))diag(z1, . . . , zn) = diag(zs(1), . . . zs(n))(the action permutes the coordinates!). Therefore, if F ∈ I(U(n)), we find that

F (diag(zs(1), . . . zs(n))) = F (Ad(ρ(s))diag(z1, . . . , zn)) = F (diag(z1, . . . , zn)).

In other words, F ∣t(n) ∈ I(T(n))Σn . The point of this argument is that theΣn-action on T(n) is the restriction of an inner action on the larger group.

Next, we recall the main theorem on symmetric polynomials [47]. It statesthat the elementary symmetric functions σ1, . . . , σn ∈ R[x1, . . . , xn]Σn are alge-braically independent and that each symmetric polynomial, i.e. each element inR[x1, . . . , xn]Σn can be written as a polynomial in the σi’s. This proves that thecomposition is an isomorphism, because it maps ck to σk(x1, . . . , xn).

To complete the proof, it remains to be shown that the restriction map I(U(n))→I(T(n)) is injective. In concrete terms, this means that an element F ∈ I(U(n)) isdetermined by its restriction to t(n). By definition, F is invariant under the adjointrepresentations: if g ∈ U(n) and X ∈ u(n), then F (gXg−1) = F (X). Now we invokethe spectral theorem from Linear Algebra (the Jordan normal form theorem forskew-hermitian matrices). It states that for each X ∈ u(n), there is g ∈ U(n) withgXg−1 a diagonal matrix, in other words gXg−1 ∈ t(n). Therefore F is determinedby its restriction to t(n).


This result is extremely useful: for many applications, it will be important todefine appropriate characteristic classes. The theorem says that for each symmetricpolynomial in R[x1, . . . , xn], we get a unique element in I(U(n)), and through theChern-Weil construction, a characteristic class for complex vector bundles. Theseclasses will of course be polynomials in the Chern classes and so do not carry moreinformation than the Chern classes itself, but they package this information in adifferent way. These arguments reduce a large portion of the theory of characteristicclasses to computations with symmetric poylnomials.

Before we do more calculations, we carry out the same argument for the otherclassical groups, i.e. SO(n) and O(n).

The orthogonal groups. The Lie algebra o(n) = so(n) of the orthogonal group is thespace of skew-symmetric matrices. The role of the subgroup T(n) will be playedby a subgroup D(n) ⊂ SO(2n). For a ∈ R, we let

Ra ∶= ( −aa

) ∈ so(2).

Let D(n) ⊂ SO(2n) be the group of all matrices

⎛⎜⎝

exp(Ra1). . .

exp(Ran)

⎞⎟⎠

with ai ∈ R. Of course, D(n) is isomorphic to T(n), but it is useful not to identifyboth groups. Its Lie algebra d(n) ⊂ so(2n) consists of all matrices of the form

A(a1, . . . , an) =⎛⎜⎝

Ra1. . .

Ran

⎞⎟⎠

with aj ∈ R. Define a linear form yj ∈ d(n)∗ by

yj(A(a1, . . . , an)) = −1

2πaj .

It is clear that I(D(n)) = R[y1, . . . , yn]. We will investigate I(O(m)) andI(SO(m)) by its restriction to D(n).

Lemma 9.3.8. For G = O(2n),O(2n+1), SO(2n) and SO(2n+1), the restrictionsI(G)→ I(D(n)) are injective.

Proof. This follows from the spectral theorem for skew-symmetric matrices. Itasserts that if X ∈ o(2n) or X ∈ o(2n + 1), we can find g ∈ O(2n) or g ∈ O(2n + 1)with gXg−1 ∈ d(n). by the same argument as the one used in the last part of theproof of Theorem 9.3.7, this implies the claim when G = O(m). If G = SO(m), weneed to find such a g which lies in SO(m). But the matrix t = diag(−1,1,1, . . . ,1)has determinant −1 and conjugates A(a1, . . . , an) into A(−a1, a2, . . . , an), and ifnecessary, we replace g by tg.

In the last term, we introduced the Euler class. The algebraic part of its con-struction can be summarized by

Proposition 9.3.9. There exists a unique e ∈ In(SO(2n)) such that e∣d(n) =y1⋯yn. The polynomial e is called Pfaffian polynomial.

150 JOHANNES EBERT

There is a homomorphism φ ∶ U(n) → SO(2n); it comes from identifying Cnwith R2n. We want to prove that φ∗e = cn; in other words, the Euler class of acomplex vector bundle, considered as an oriented real vector bundle, is equal to thetop Chern class. This depends on an orientation convention, or on the choice ofCn ≅ R2n; we use

(z1, . . . , zn)↦ (R(z1),I(z1), . . . ,R(zn),I(zn)).This gives rise to φ ∶ U(n)→ SO(2n), and φ(T(n)) = D(n). More precisely

φ(diag(ia1, . . . , ian)) = A(a1, . . . , an),whence

φ∗yj = xjand hence

Proposition 9.3.10. Under the homomorphism φ ∶ U(n)→ SO(2n), φ∗e = cn. Inother words, for each complex rank n vector bundle V →M , we have e(V ) = cn(V ).

Theorem 9.3.11.

(1) e(V ⊕W ) = e(V )e(W ).(2) If a rank n complex vector bundle V →M is considered as a real oriented

vector bundle, then e(V ) = cn(V ). The orientation convention is that if(v1, . . . , vn) is a C-basis, then (v1, iv1, . . . , vn, ivn) is an oriented R-basis.

(3) Reversing the orientation reverses the sign of e.

Proof. The second part follows from Proposition 9.3.10. We proved the other twoparts last term. Part (1) follows from the formula given in 9.3.9, together with9.3.8. The proof given for the third part used the Gauß-Bonnet-Chern theorem,and therefore we indicate an algebraic proof. The point is that conjugation with thematrix t = diag(−1,1,1, . . . ,1) induces an automorphism ι of so(2m) which leavesd(n) invariant. Moreover, ι∗y1 = −y1 and ι∗yj = yj for j > 1 so that ι∗e = −e. Butchanging an SO(2m)-principal bundle by ι corresponds to reversing the orientationof a vector bundle.

The Gauss-Bonnet-Chern theorem mentioned in the proof was a very substantialresult. We can rephrase it as follows. Let V →M be an oriented vector bundle ofrank k over an oriented closed manifold. Let s be a section of V that is transverseto the zero section, and let Z be the zero set of s. Then the Euler class is Poincaredual to Z, in other words

∫Mω ∧ e(V ) = ∫

Zω

holds for each closed form ω on M . Moreover, if V = TM , we derived that

∫Me(TM) = χ(M).

There is also a homomorphism ψ ∶ O(m)→ U(m) (not depending on any choice,it is just an inclusion). On the vector bundle side, ψ corresponds to taking thecomplexification VC = V ⊗R C of a real vector bundle. We wish to compute ψ∗ck.

For odd values of k, this is easy. Note that ψ(A) = ψ(A) and hence, due to 9.3.5,we have c2k+1(ψ(A)) = −c2k+1(ψ(A)) and therefore


ψ∗c2k+1 = 0 ∈ I2k+1(O(m)).

Definition 9.3.12. The Pontrjagin polynomial is the element pk ∶= (−1)kψ∗c2k ∈I2k(O(m)). The total Pontrjagin polynomial is p = ∑k≥0 t

kpk ∈ I(O(m))[t]. Usingthe Chern-Weil construction, we define the Pontrjagin class of a rank m real vectorbundle pk(V ) ∈ H4k(M ;R) and the total Pontrjagin class p(V ) = 1 + p1(V ) + . . . +pn(V ), where n = ⌊m/2⌋ is the largest integer with 2n ≤m.

Let n = ⌊m/2⌋ (i.e. m = 2n or m = 2n + 1). We wish to derive a formula forthe Pontrjagin polynomial. The key observation is that the eigenvalues of a matrixA ∈ o(n) are purely imaginary and occur in complex conjugate pairs, i.e. theeigenvalues of A ∈ so(2n) are iµ1,−iµ1, . . . , iµn,−iµn; if A ∈ o(2n+ 1), we have 0 asthe last eigenvalue. We find that

∑k≥0

(−1)kpk(A) =n

∏j=1

(1 + t(− 1

2πiiµj)(1 − t(−

1

2πiiµj) =

n

∏j=1

(1 − t2( 1

(2π)2µ2j).

If A ∈ d(n) ⊂ o(m), we find that 12πµj = ±yj(A), and so we get that

(9.3.13) pk ∣d(n) = σk(y21 , . . . , y

2n).

In the same way as for Chern classes, we obtain that

p((AB)) = p(A)p(B).

A special case is when B = 0, in this case the above equation simply says thatpk ∣o(n) = pk. An immediate consequence of 9.3.13 is

e2(A) = pn(A)holds for matrices in d(n), and by Lemma 9.3.8, we get that e2 = pn ∈ I(SO(2n)).

To summarize the discussion so far, we have homomorphisms

R[p1, . . . , pn]→ I(O(2n + 1))R[p1, . . . , pn]→ I(O(2n))

R[p1, . . . , pn]→ I(SO(2n + 1))R[p1, . . . , pn, e]/(e2 − pn)→ I(SO(2n)).

(9.3.14)

Theorem 9.3.15. The homomorphisms 9.3.14 are isomorphisms.

Proof. We use the maps

SO(2n)→ O(2n)→ SO(2n + 1)→ O(2n + 1),

the map O(2n)→ SO(2n+ 1) sends A↦ (Adet(A)), and the other two maps

are the inclusions. The composition SO(2n) → SO(2n + 1) is the usual inclusion(but O(2n) → O(2n + 1) is slightly different). When composed with the inclusionD(n) ⊂ SO(2n), all the inclusions of D(n) are the usual ones. Now look at thediagram

152 JOHANNES EBERT

R[p1, . . . , pn]

vv ''I(O(2n + 1)) //

((

I(SO(2n + 1)) //

I(O(2n))

wwI(D(n))

The arrows ending in in I(D(n)) are all injective by Lemma 9.3.8 and the lowertriangles commute. So all horizontal homomorphisms are injective. All homomor-phism from the top to the bottom agree and are injective. So the upper trianglescommute as well. If we can show that R[p1, . . . , pn] → I(O(2n)) is surjective,a diagram chase shows that the upper vertical arrows are all isomorphisms. LetW ⊂ O(2n) be the following finite subgroup. It is generated by the image ofthe permutation representation Σn → U(n) ⊂ O(2n), together with the elementt = diag(−1,1,1, . . . ,1). It can be shown without difficulty that W ≅ Σn ⋉ (Z/2)n.The group W acts on D(n) by conjugation. On I(D(n)), it acts by (s ∈ Σn)

t∗y1 = −y1; t∗yj = yj ; s∗yj = ys(j).Let F ∈ I(O(2n)) and w ∈W . Then w∗F = F (F is invariant under conjugation

by O(2n) and hence by W ⊂ O(2n)) and we have maps

R[p1, . . . , pn]→ I(O(2n))→ I(D(n))W .Both maps are injective, and if we can show that the composition is surjective,

then both maps are isomorphisms, and the proof is complete. Let F (y1, . . . , yn) ∈I(D(n))W . What we have to show that it is a polynomial in the σk(y2

1 , . . . , y2n).

For each j, let sj ∈ Σn be the transposition (1j). The element tj ∶= sjtsj ∈W actson the generators of I(D(n)) by

t∗j yk = (−1)δjkyk.Invariance of F under W therefore implies that F is even in each of its variables.

Therefore, F ∈ R[y21 , . . . , y

2n]. But invariance of F under Σn ⊂ W means that F is

symmetric, in other words, F is a polynomial in the σk(y21 , . . . , y

2n).

It remains to study the case SO(2n). There are still maps

R[p1, . . . , pn, e]/(e2 − pn)→ I(SO(2n))→ I(D(n)).The second one is injective by Lemma 9.3.8, and since we can write

R[p1, . . . , pn, e]/(e2 − pn) = R[p1, . . . , e],the composition is still injective. We can no longer argue that the image of

I(SO(2n)) is invariant under W . However, for w ∈W , we have Ad(w)e = det(w)e.Let V be the kernel of det ∶ W → ±1. It is V = W ∩ SO(2n), and thereforeI(SO(2n)) → I(D(n))V . It remains to show that R[p1, . . . , pn, e]/(e2 − pn) →I(D(n))V is surjective. Let ι ∶= Ad(t) ∶ I(D(n))V → I(D(n))V . This is an auto-morphism of order 2. If ιF = F , then F is invariant under W and hence, by whatwe showed before, a polynomial in p1, . . . , pn. If ιF = −F , then F (−y1, y2, . . . , yn) =−F (y1, . . . , yn). By the Σn-invariance, we find that F is odd in each variable yj .


Hence F is divisible by yj and hence by e = y1⋯yn. Thus we can write F = eG,and G satisfies ιG = G. Therefore G is a polynomial in the Pontrjagin polynomials.This completes the proof.

9.4. Maximal tori in compact Lie groups. The next major goal is a proof ofthe following result by H. Cartan.

Theorem 9.4.1. For each compact Lie group G, the universal Chern-Weil mapI(G)→H∗(BG;R) is an isomorphism (it is part of the statement that H2∗+1(BG;R) =0).

We already proved this result for G = T(n) (Theorem 9.3.2), and the generalcase will be done by reduction to this case. The reduction argument requires one ofthe central results of the theory of compact Lie groups, namely the maximal torustheorem. This generalizes the Jordan normal theorem to arbitrary compact Liegroups. It is often used in Lie theory, for a similar purpose. I must admit two points:one step in the proof of Theorem 9.4.1 is the analogue for the invariant-theoreticTheorems 9.3.15 and 9.3.7. I do not know a simple argument for the general case,and I will simply quote the result. Also, one can circumvent the use of the generalmaximal torus theorem when one is only interested in the classical groups. However,this requires to compute the Euler number of the homogeneous space Gr+2(R2n),and I also do not know a simple argument for that either. Moreover, there is a clearmotivation for giving the proof here, since it is one the most substantial applicationsof one of the results of the previous term: the Poincare-Hopf theorem. We will also( but mostly to make the proofs shorter) use a close corollary to the Peter-Weyltheorem:

Theorem 9.4.2. For each compact Lie group G, there is an injective homomor-phism G→ U(n), for some n.

The proof of the Peter-Weyl theorem requires the spectral theorem for compactself-adjoint operators on a Hilbert space.

Definition 9.4.3. A torus is a Lie group which is isomorphic to T(n) ≅ Rn/Zn. Amaximal torus in a compact Lie group G is a torus T ⊂ G such that if T ⊂ S ⊂ Gis another torus, then S = T . The Weyl group W of G (with respect to T ) is thequotient NGT /T of the normalizer of T by T .

Theorem 9.4.4. ([21]) A compact connected abelian Lie group is isomorphic to atorus.

An important feature of tori is that they are topologically cyclic.

Theorem 9.4.5. (Kronecker) Let T be a torus. Then there exist an element x ∈ Tsuch that the subgroup ⟨x⟩ generated by x is dense in T . Moreover, there is anelement X ∈ t such that the subgroup ⟨exp(tX)∣t ∈ R⟩ is dense in T .

Proof. By the previous theorem, we can take T = Rn/Zn. Let Σ ⊂ T be the union ofall kernels of nontrivial homomorphisms T → T. Since homomorphisms correspondto integral linear forms Rn → R, there are countably many such homomorphisms.Therefore, Σ has Lebesgue measure zero, and so T ∖ Σ is nonempty. If x ∈ T and

S ∶= ⟨x⟩ ≠ T , then T /S is, by the previous theorem, a torus of positive dimension.Thus there is a nontrivial homomorphism f ∶ T → T such that f(x) = 1, and x ∈ Σ.This finishes the proof of the first part, and the second follows easily.

154 JOHANNES EBERT

Remark 9.4.6. One can show without pain that if aj ∈ R, j = 1, . . . , n and if(1, a1, . . . , an) is linearly independent over Q, then the element [a1, . . . , an] ∈ Rn/Znis a topological generator.

Proposition 9.4.7. Let G be a connected compact Lie group and T ⊂ G be amaximal torus. The normalizer NT of T in G is compact and the Weyl groupW = NT /T is finite.

Proof. Since NT = ∩t∈T g ∈ G∣gtg−1 ∈ T, it is closed in G, hence a Lie groupand compact. The normalizer NT acts on T by conjugation. This yields a grouphomomorphism NT → Aut(T ). The automorphism group Aut(T ) is isomorphic toGLn(Z), n = dim(T ) and therefore discrete. Thus NT0, the unit component of NT ,acts trivially on T . Let X ∈ Lie(NT ). Then exp(sX)t exp(−sX) = t for all t ∈ Tand s ∈ R. Let S be the closure (in NT ) of the group generated by exp(sX), s ∈ Rand T . The group S is abelian and compact, hence a torus, and T ⊂ S. Since T ismaximal, S = T . This argument proves that NT0 = T . Therefore, NT /T ≅ π0(NT ),and this is finite.

Theorem 9.4.8. (The maximal torus theorem) Let G be a compact connected Liegroup and T ⊂ G be a maximal torus.

(1) The Euler number of the homogeneous space G/T is equal to ∣W ∣, the orderof the Weyl group (Hopf-Samelson).

(2) If S ⊂ G is another maximal torus, there exists g ∈ G with gSg−1 = T .(3) ⋃g∈G gTg−1 = G.

Proof. For each X ∈ g, we get a vector field νX on G/T , by taking the derivative ofthe translation action by exp(tX);

νX(gT ) = d

dt∣t=0(exp(tX)gT ) ∈ TgTG/T.

For X ∈ g, denote ⟨X⟩ ∶= exp(TX)∣t ∈ R. Now observe the following equivalences:

νX(gT ) = 0 exp(sX)gT = gT for all s ∈ R g−1 exp(sX)g ∈ T for all s ∈ R g−1⟨X⟩g ⊂ T .

Now assume that part 1 is shown. Part 2 is derived as follows: let S ⊂ Gbe a torus. Choose X ∈ s with ⟨X⟩ = S, by Kroneckers theorem 9.4.5. By thePoincare-Hopf theorem and the first statement, νX has a zero gT . Then g−1Sg ⊂ T ,as desired. The third part is derived in a similar way, but not necessary for ourpurposes. One uses the action maps Lg ∶ G/T → G/T and shows that they allhave fixed points. One needs the Lefschetz fixed point theorem and the fact thatχ(G/T ) ≠ 0.

To prove the first part, choose X ∈ t so that ⟨X⟩ = T . By the equivalencesdisplayed above, the zeroes of νX are precisely the points gT with g ∈ NT , in otherwords the points of the Weyl group. Lemma 9.4.9 below implies that the localindices of νX are all +1, so that χ(G/T ) = ∣W ∣, by Poincare-Hopf. We need tojustify the use of the lemma. Using the Haar measure on G, one produces a G-invariant metric on G/T . The flow of the vector field νX is the map R×G/T → G/T ,(s, gT )↦ exp(sX)gT , and this is by isometries.


Lemma 9.4.9. Let M be a compact Riemann manifold and X a vector field withisolated zeroes, such that the flow Φt generated by X acts by isometries on M . Thenthe vector field X is transverse to the zero section and the local index of X at everyzero is +1.

Proof. Let x be a zero of X. Let A ∈ End(TxM) be the derivative of X at x,

considered as a map TxDxX→ T(x,0)TM = Tx ⊕ Tx → TxM , the last map is the

projection to the second coordinate. We have to prove that det(A) > 0.The flow map Φ ∶ R×M →M is smooth, and we consider the evolution equation

for the derivative DxΦt at x. The derivative is a function Z ∶ R → GL(TxM), andit satisfies the ODE

Z(0) = 1;d

dtZ(t) = AZ(t).

Therefore Z(t) = exp(tA). For v ∈ TxM , consider the geodesic γ emanating fromx with initial speed v. Since the flow is by isometries, φt γ is again a geodesic,emanating from φt(x) = x and with initial speed exp(tA)v. If Av = 0, then itfollows that φt γ = γ, and so φt fixes all the points γ(s), s ∈ R. Thus X(γ(s)) = 0,and if v ≠ 0, this is a contradiction to the assumption that X has isolated zeroes.Therefore det(A) ≠ 0.

Since Φt is an isometry, the automorphism Z(t) is orthogonal, and so A is skew-symmetric, and therefore its spectrum is ±µ1 . . . ,±µn, with real µj ≠ 0. Thereforedet(A) =∏n

j=1 µ2j > 0.

Definition 9.4.10. The rank of the Lie group G is the dimension of a maximaltorus in G (well-defined by the maxial torus theorem).

Corollary 9.4.11. For each connected compact Lie group, the inclusion T → G ofa maximal torus induces an injection I(G)→ I(T )W .

Proof. Let F ∈ I(G). Let X ∈ t and w ∈ W . Pick an element g ∈ NT whichrepresents w ∈ NT /T . Then, by the invariance of F :

F (wX) = F (Ad(g)X) = F (X),and this shows that I(G) → I(T )W . For the injectivity, assume that F ∣t = 0 and

let X ∈ g. Consider the torus S = exp(RX) ⊂ G. By the maximal torus theorem,there is g ∈ G with gSg−1 ⊂ T . Hence Ad(g)X ∈ t and F (X) = F (Ad(g)X) = 0.

Theorem 9.4.12. For each connected Lie group G, the restriction map I(G) →I(T )W is an isomorphism. The algebra I(T )W is isomorphic to a polynomial al-gebra on r generators, where r = dim(T ) is the rank of G.

We proved this for G = U(n);SO(n). For a general compact Lie group, this is adifficult theorem due to Chevalley. Proofs can be found in [23] or [39].

9.5. Proof of Cartan’s theorem. The proof of Cartans theorem will be in twosteps, both of which are similar. Assume that G is connected. The diagram

I(G) CW //

H∗(BG)

I(T )W CW // H∗(BT )W

156 JOHANNES EBERT

exists by Lemma 9.2.2 and is commutative. We have seen that the bottomhorizontal map is an isomorphism in Theorem 9.3.2 and that the left vertical mapis an isomorphism, for all groups we are interested in. Thus the proof is completeif we can show that H∗(BG;R)→H∗(BT ;R) is injective.

If G is not connected, let K ⊂ G be the unit component and Γ = π0(G) = G/K bethe component group. The group Γ acts by conjugation on K, and thus on I(K).We have a similar commutative diagram

I(G) CW //

H∗(BG)

I(K)Γ CW // H∗(BK)Γ.

We assume that we have shown that the bottom horizontal map is an isomor-phism. It is quite obvious that the left vertical map is an isomorphism (k ≅ g). Ifwe can show that H∗(BG;R)→H∗(BK;R) is injective, the proof is complete.

In both cases, we need to show that BH → BG is injective, where H ⊂ G isa closed subgroup. The common property (which is also the key to the proof) isthat the Euler number χ(G/H) is nonzero; for T ⊂ G, this is the Hopf-Samelsontheorem, and for K ⊂ G, this is clear. A model for BH is the space EG/H, andthus the fibre bundle

EG ×G G/H → BG

is a model for BH → BG. By the argument using the connectivity of the Stiefelmanifold and the Hurewicz theorem, it is enough to prove that for each k, the mapq ∶ EkG×GG/H → BkG induces an injection on real cohomology. Both, source andtarget of q are closed manifolds. An orientability problem requires to treat bothcases separately, however.

Theorem 9.5.1. Let q ∶ E → M be a fibre bundle of closed manifolds, let Mbe connected and assume that the Euler number of F = q−1(x) is nonzero. Thenq∗ ∶H∗(M ;R)→H∗(E,R) is injective.

Proof. One case to consider is when q is a finite covering with n sheets. We definethe transfer trfq ∶ A∗(E) → A∗(M) as follows. Let η ∈ A∗(E). Let U ⊂ M bean open set over which the convering is trivial and let Uj , j = 1, . . . , n be thecomponents of q−1(U). Define

(trfq(η))∣U ∶=n

∑j=1

((q∣Uj)−1)∗η ∈ A∗(U).

It is easy to see that these locally defined differential forms fit together to aglobally defined form trfq(η). Moreover, trfq ∶ A∗(E)→ A∗(M) is a chain map andtrfq(q∗η) = nη. It follows that on cohomology, we get

trfq q∗ = nid.

Thus q∗ is injective (since n ≠ 0). In the general case, let M → M be the

orientation cover and E → M be the pullback of the bundle q. The diagram


H∗(M) //

q∗

H∗(M)

q∗

H∗(E) // H∗(E)

proves that it is enough to show the theorem when the base manifold is oriented(the horizontal maps are injective by the first part of the proof). So let M be

oriented. By passing to the orientation cover E → E, we can, by the same argument,assume that E is oriented as well. So we assume that E and M are oriented, and letk ∶= dimE−dimM = dim q−1(x). The vertical tangent bundle is TvE ∶= ker(dq)→ E.There is a splitting TE ≅ TvE ⊕ q∗TM . The restriction of the vertical tangentbundle to any fibre q−1(x) is just Tq−1(x). By the orientation convention, TvEinherits an orientation from those of M and E, so that we can talk about the Eulerclass e(TvE) ∈ Hk(E;R). We also have the Gysin homomorphism q! ∶ H∗(E) →H∗−k(M). We define the transfer trfq ∶H∗(E)→H∗(M) (no degree shift) by

trfq(x) = q!(e(TvE)x).Now, the well-known property of the Gysin map and the topological Gauss-

Bonnet theorem yields

trfq(q∗x) = q!(e(TvE)q∗x) = q!(e(TvE))x = χ(F )x.As we assumed that χ(F ) ≠ 0, the proof is complete.

9.6. Examples. We now will explicitly compute some characteristic classes, for thetangent bundle of some manifolds. Some of the computations will go into the proofof the index theorem, and the purpose of the other computations is to show thatthese invariants are computable in some highly nontrivial cases. You will notice thefeature that no really new computation is necessary, the only explicit calculationwas that of the first Chern class of the tautological bundle on CPn. One way to viewthese calculations is to understand them as methods to compute certain integralsexplicitly using topology. The greatest virtuoso of these calculations was withoutdoubt Friedrich Hirzebruch, and it is a safe conjecture to say that without thesecomputations, the general formulation of the index would not have been found.

Stably parallelizable manifolds.

Definition 9.6.1. A vector bundle V → M is stably trivial if there exists k suchthat V ⊕Rk is trivial. A manifold M is stably parallelizable if TM is stably trivial.

If TM is stably trivial and TM ⊕ Rk is trivial, then in fact TM ⊕ R is trivial.Typical examples of stably parallelizable manifolds, besides parallelizable mani-folds, are oriented hypersurfaces Mn ⊂ Rn+1. The reason is that the normal bundleof the inclusion M ⊂ Rn+1 is an oriented real line bundle and hence trivial, so thatTM ⊕ νM = TM ⊕R is trivial. Now we see that

pk(V ⊕R) = pk(V )and thus if M is stably parallelizable, then pk(TM) = 0 for all k > 0. An

example includes Sn. Therefore, all Pontrjagin classes of TSn are trivial. Boringas it sounds, this fact has a surprising consequence.

158 JOHANNES EBERT

Theorem 9.6.2. The vector bundle TS4n does not admit the structure of a complexvector bundle, for n > 0. Thus S4k is not a complex manifold.

Proof. Assume that TS4n is a complex vector bundle. Then

0 = pn(TS4n) = (−1)nc2n(TS4 ⊗R C) = (−1)nc2n(TS4n ⊕ TS4n) =

(−1)n(c2n(TS4n) + c2n(TS4n)) = (−1)n2c2n(TS4n) = 2(−1)ne(TS4n).

The first equation is that S4n is stably parallelizable, the second is the definitionof the Pontrjagin class. For the third, use that V ⊗R C = V ⊕ V if V is a complexvector bundle. The fourth equation is the product formula for Chern classes (thelower Chern classes are zero, since Hk(S4n) = 0 for 0 > k > 4n). The last equationis the relation between Euler and Chern classes. But this is a contradiction: by theGauß-Bonnet theorem, e(TS4n) is nonzero; it integrates to 2 = χ(S4n).

Using Bott periodicity, we will prove below that S2n cannot have a complexstructure if 2n ≥ 8.

Complex projective spaces. For further computations, we need to know the tangentbundle of CPn.

Theorem 9.6.3. Let V be a finite-dimensional complex vector space and let LV ,HV →PV be the tautological line bundle and its dual. Then there is an isomorphismTPV ≅ Hom(LV , V )/C; C = Hom(LV , LV ) ⊂ Hom(LV , V ).

Proof. Let Φ ∶ End(V )→ TPV be given by

(`, f)↦ d

dt∣t=0(etf `) ∈ T`PV

and Ψ ∶ End(V )→ TPV by

End(V )∣LV→ Hom(LV ;V )→ Hom(LV ;V )/Hom(LV , LV ).

These are two bundle maps, and it is clear that Ψ is an epimorphism. It is clearthat ψ is complex linear, and it is crucial that φ is C-linear as well. This requiresa little thought. The point is that GL(V ) is a complex Lie group and that theaction on PV is holomorphic. Moreover, the targets of both maps have the samedimension. Let (`, f) ∈ ker(Φ). Then etf ` = `. This means that ` ⊂ V is f -invariant;and therefore Ψ(`, f) = 0. In other words, ker(Φ) ⊂ ker(Ψ). For dimension reasons,this shows that Φ is surjective as well. Since both kernels are the same, it followsthat

Hom(LV ;V )/Hom(LV , LV ) ≅ TPV,as claimed.

I believe that the invariant language used in the proof makes the argument moretransparent. Using the fact that exact sequences split, we arrive at TPV ⊕ C ≅Hom(LV , V ). This determines TPV as an element in K0(PV ). When V = Cn+1,we rewrite this isomorphism as

TCPn ⊕C ≅H⊕(n+1).


This is enough to compute all characteristic classes. Let x = c1(H) ∈ H2(CPn).We find that

c(TCPn) = c(TCPn ⊕C) = c(H)n+1 = (1 + x)n+1.

Moreover

e(TCPn) = cn(TCPn) = (n + 1)xn.Together with ∫CPn x

n = 1, this confirms that χ(TCPn) = (n+1), using the Gauß-Bonnet theorem. It is at least psychologically important to have such calculationsto check signs. The Pontrjagin classes can be computed by

(9.6.4) p1(H) = −c2(H ⊗R C) = −c2(H ⊕ H) = −c1(H)c1(H) = c1(H)2 = x2

and

p(TCPn) = p(TCPn ⊕C) = p(H)n+1 = (1 + x2)n+1.

For example, this calculation shows that CPn is not stably parallelizable unlessn = 1.

Algebraic hypersurfaces. A very interesting class of examples are the projective hy-persurfaces in CPn. Let V be a finite-dimensional vector space and let d ≥ 0.Consider a homogeneous polynomial q ∈ Pold(V ) of degree d on V . We obtain across-section sq of H⊗d

V → PV . Its value at ` ∈ PV is given by the restriction q∣`,which is a degree d homogeneous polynomial on `, hence an element of (`∗)⊗d. Infact, sq is a holomorphic section and one can show that all holomorphic sections of

H⊗dV are of this form. Let V = Cn+1. If q is such that sq ⋔ 0, then Vq ∶= s−1

q (0) ⊂ CPnis a complex submanifold of codimension 1.

It is not entirely clear that there exists a polynomial q with sq ⋔ 0. This is aninstance of Bertini’s theorem and the present version follows from Sards theorem(we will not prove it). Let S ⊂ Pold(V ) be the subset of polynomials which are

not transverse. Then S is an algebraic subset, and since S ≠ Pold(V ), S has realcodimension at least 2. Therefore V ∖ S is open and path-connected. This can beused to prove that the diffeomorphism type of Vq only depends on d (you mightfind it an interesting problem to give a rigorous proof of this fact. I have twohints: for a path qt of regular polynomials, consider W = ∪t∈[0,1]Vqt , prove thatit is a manifold and that the projection to [0,1] has no critical points. We willtake these things for granted and use the notation Vd. Another important fact isthe Lefschetz hyperplane theorem. It states that the inclusion map Vd → CPn is(2n − 2)-connected. The proof can be given using Morse theory [50].

Now we calculate the characteristic classes of V ∶= Vd,n ⊂ CPn+1. The key is that

the normal bundle of Vd,n ⊂ CPn+1 is just H⊗d∣V . Therefore

TV ⊕H⊗d∣V ⊕C ≅ TCPn+1∣V ⊕C ≅H⊕(n+2)∣V .Let z ∶= x∣V ∈H2(V ). We obtain that

c(TV )(1 + dz) = c(TV )c(H⊗d∣V ) = c(H ∣V )n+2 = (1 + z)n+2

and, using 9.6.4,

160 JOHANNES EBERT

p(TV )(1 + d2z2) = p(H ∣V )n+2 = (1 + z2)n+2.

Furthermore, writing j ∶ V → CPn+1 and using Theorem 6.3.7, we see that

∫Vj∗xn = ∫

CPnxne(H⊗d) = ∫

CPnxndx = d.

The term (1 + dz) is invertible in H∗(V ), with (1 + dz)−1 = ∑k≥0(−1)kdkzk.Therefore, cn(TV ) is the nth term of the polynomial ∑nk=0(−1)kdkzk(1+z)n+2. Forn = 1, we compute

c(TVd,1) = (1 + 3z)(1 − dz) = 1 + (3 − d)zUsing the formula for the integral and the relation χ = 2 − 2g, we get

g(Vd,1) = 1 + d2(d − 3),

the degree-genus-formula (the low values match with well-known classical resultsfrom the theory of algebraic curves).

For n = 2, we find

c(TVd,2) = (1 + z)4(1 − dz + d2z2) = 1 + (4 − d)z + (6 − 4d + d2)z2

and by the same reasoning as for n = 1:

χ(TVd,2) = d(6 − 4d + d2).If d = 1, this reaffirms the known computations for CP2. The case d = 4 is

interesting: the first Chern class is zero, and the Euler number 24. This 4-manifoldis called the K3-surface (your search for the 3 in the numerology will be in vain:they were named by Weil to honour the three mathematicians Kahler, Kodairaand Kummer, and the name is also an allusion to the peak K2 in Pakistan). ThePontrjagin classes are

p(TV ) = (1 − d2z2)(1 + 4z2) = 1 + (4 − d2)z2

and hence

p1[V ] ∶= ∫Vp1(TV ) = d(4 − d2).

Why is this numerology interesting? Well,

p1[V ] = d(4 − d2) ≡ d(1 − d2) = −d(d − 1)(d + 1) ≡ 0 (mod 3).In fact, we will ultimately prove that sign(M4) = ± 1

3p1[M] (the signature theo-

rem). Even more interesting, if d is even, then c1(TV ) is an even multiple of z. Inthis case, if d = 2r, we find

p1[V ] = 2r(4 − 4r2) = −8r(r − 1)(r + 1).Since r(r − 1)(r + 1) is divisible by 6, we find that for d even, p1[V ] is divisible

by 48. In fact, we will prove that d even implies that V is ”spin”, and that thereis an elliptic operator /D with ind( /D) = ± 1

24p1[V 4]. There is a factor of 2 missing:

well, the index of /D will be even.


9.6.1. Quaternionic projective spaces. Just as in the real or complex case, we candefine quaternionic projective space HPn. It is the space of all 1-dimensional quater-nionic subspaces of Hn+1. Recall that S3 ⊂ H is the group of norm 1 quaternions.Let Q = S4n+3 ⊂ Hn+1. There is a right action of S3 on Q by multiplication; thequotient Q/S3 is - by definition - HPn and Q → HPn is an S3-principal bundle.Observe that S1 = R⊕ iR∩S3. The group S3 acts by conjugation on the imaginarypart IH ≅ R3; this yields a homomorphism S3 → SO(3); just another incarnationof the spin covering. Let V → HPn be the vector bundle Q ×S3 V → HPn and letw ∶= p1(V ) ∈H4(HPn). The sphere bundle in V is just

Q ×S3 S2 = Q/S1 = CP2n+1,

and the projection map q ∶ CP2n+1 → HPn has an alternative description: it takesa complex line in Hn+1 = C2n+2 to its right H-span. Since V is 3-dimensional, itsEuler class is zero, and so the Gysin sequence splits into short exact sequences:

0→Hk(HPn) q∗

→Hk(CP2n+1) q!→Hk−2(CP2n+1)→ 0.

In particular, the map q∗ is injective. A calculation of representations provesthat the restriction of the representation S3 IH to S1 splits as a direct sumR ⊕ A2, where R has the trivial action and A2 = C the action by z2 (the sign isirrelevant: the only characteristic class that is sensitive to orientation is the Eulerclass, which is zero). Therefore, q∗V is the vector bundle R⊕H⊗±2, with Pontrjaginclass 4x2 (as usual x = c1(H)). Thus q∗w = 4x2. The factor 4 should alarm you. Infact, there is a better choice for a basic class in H4(HPn). But we can determinethe additive structure of H∗(HPn). The above sequence proves that in odd degrees,there is no cohomology. Furthermore, we have seen that H4(HPn) is nontrivial (itcontains w). Since H2k(CP2n+1) is (at most) onedimensional, we find that exactlyone of the groups H2k(HPn) and H2k+2(HPn) is nonzero (and 1-dimensional). ThatH4(HPn) ≠ 1 proves that H2(HPn) = 0. Therefore H4i(HPn) = R, i = 0, . . . , n, andthe other groups are null.

To proceed, we consider the natural left-action of S3 on H. This gives a quater-nionic line bundle M → HPn. We can forget the quaternionic structure and onlykeep the complex structure. Let u = −c2(M) ∈ H4(HPn). One can compute, againusing representation theory, that q∗M = H ⊕ H. Therefore q∗u = x2 and u = 1

4w.

Compute (using the same trick we used many times)

∫HPn

un = ∫CP2n+1

q∗unx = ∫CP2n+1

x2n+1 = 1.

We conclude that H∗(HPn) = R[u]/(un+1). There is a splitting of bundles

TvCP2n+1 ⊕ q∗THPn = TCP2n+1.

The vertical tangent bundle TvCP2n+1 is H⊗2. Therefore

(1 + 4x2)q∗p(THPn) = (1 + x2)2n+2

or, since q∗ is injective,

p(THPn) = (1 + x2)n+1(1 + 4x2)−1 = (1 + u)2n+2(1 + 4u)−1.

For n = 1, we obtain p(THP1) = p(TS4) = 1 (reality check!). If n = 2, we get

162 JOHANNES EBERT

p(THP2) = 1 + 2u + 7u2.

[The formula for the L- and A-class, together with the knowledge that A = 0 andsign = 1 shows that this result is correct]

9.7. The Chern character. There is an important connection between charac-teristic classes and K-theory, given by the Chern character that we introduce now.We write

I(G) =∏k≥0

Ik(G)

as opposed to I(G) = ⊕k≥0 Ik(G). We can apply the Chern-Weil constructionwith elements in this ”completed” ring (since H∗(M ;R) = 0 for ∗ > dim(M)).

Definition 9.7.1. The Chern character form ch ∈ I(U(n)) is given by

ch =n

∑i=1

exi =n

∑i=1

∑k≥0

1

k!xki ∈ R[[x1, . . . , xn]]Σn = I(U(n)).

Via the Chern-Weil construction, we define the Chern character of the vectorbundle V →M by

ch(V ) ∶= CW(Fr(V ), ch) ∈H∗(M ;R).

The Chern character form can be written, using the polynomial

sk =k

∑i=1

xki ∈ Ik(U(n)),

as

ch = n +∞∑k=1

1

k!sk.

Alternatively

ch(A) = Tr(exp(− 1

2πiA)).

It is easy to see that for X ∈ u(n); Y ∈ u(m), we have

(9.7.2) ch((XY)) = ch(X) + ch(Y ),

and this identity implies that for two vector bundles V,W →M :

(9.7.3) ch(V ⊕W ) = ch(V ) + ch(W ).

Moreover, we consider the Lie group homomorphism φ ∶ U(n)×U(m)→ U(mn),(A,B)↦ A⊗B. Its derivative is the function

ϕ ∶ u(n) × u(m)→ u(mn); (X,Y )↦X ⊗ 1 + 1⊗ Y.Let us compute


ch(X ⊗ 1 + 1⊗ Y ) = Tr(exp( −1

2πi(X ⊗ 1 + 1⊗ Y ))).

Since 1⊗ Y and X ⊗ 1 commute, we get

exp( −1

2πi(X ⊗ 1 + 1⊗ Y )) = exp( −1

2πi(X ⊗ 1)) exp( −1

2πi(1⊗ Y )) =

(exp( −1

2πiX)⊗ 1)(1⊗ exp( −1

2πiY ) = exp( −1

2πiX)⊗ exp( −1

2πiY ).

The equation Tr(A⊗B) = Tr(A)Tr(B) is obvious, and therefore

(9.7.4) ch(X ⊗ 1 + 1⊗ Y ) = ch(X)ch(Y )which implies

(9.7.5) ch(V ⊗W ) = ch(V )ch(W )for two vector bundles V,W → M . Now let M be a compact manifold. The

equation 9.7.3 tells us that V ↦ ch(V ) defines a group homomorphism

K0(M)→H2∗(M ;R)which by 9.7.5 is even a ring homomorphism. We can extend the definition of the

Chern character to each compact space, using the classification of vector bundles.Namely, let V → X be a vector bundle over a compact space. There is a mapfV ∶ X → Grn(Cm) with f∗V Vn,m ≅ V . The space Grn(Cm) is a compact manifold(compactness is not really necessary here), and the tautological vector bundle issmooth. Therefore, we get ch(Vn,m) ∈ H2∗(Grn,m(Cn)). Now recall the de Rhamisomorphism

H∗dR(M)→H∗

sing(M ;R).Using the de Rham homomorphism, we get a class ch(Vn,m) ∈H2∗

sing(Grn(Cm)),which we can pull back using fV :

ch(V ) ∶)f∗V ch(Vn,m) ∈H2∗sing(X;R).

By the homotopy uniqueness of the map fV , this definition of the Chern characteris unambigous. A simple argument involving classifying maps (an exercise) showsthat 9.7.3 and 9.7.5 continue to hold. Therefore, for each compact space X, weobtain a ring homomorphism

ch ∶K0(X)→H2∗(X;R).The Chern character can be extended to locally compact spaces: define H∗(X,R) ∶=

ker(H∗(X;R) → H∗(x,R)) and H∗c (X) ∶= H∗(X+;R) (it is not hard to show that

this definition coincides with the previously given definition of compactly supportedcohomology). Since the Chern character is natural, we obtain

K0c (X)→H2∗

c (X;R).The multiplicativitiy of the Chern character leads to the commutativity of the

diagram

164 JOHANNES EBERT

(9.7.6) K0c (X) ×K0

c (Y )

ch

// K0c (X × Y )

ch

H2∗c (X;R) ×H2∗

c (Y ;R) // H∗2c (X × Y )

(the bottom horizontal map is the cross product in cohomology). It should bepretty clear that we need to know the value of the Chern character on the Bott class.This computation is fairly easy: Namely, consider H → CPn. Then (x = c1(H)).

ch(H) = ec1(H) = ex = 1 + x + 1

2x2 + . . . + 1

n!xn ∈H∗(CPn).

Recall that the Bott class b ∈Kc(R2) corresponds to H −1 ∈ K0(CP1) under the

isomorphism Kc(R2) ≅ K0(CP1). From

∫CP1

ch(H − 1) = ∫CP1

1 + x − 1 = 1,

we obtain

∫R2

ch(b) = 1.

Using the commutativity of 9.7.6, we arrive at the following result.

Theorem 9.7.7. For each n ≥ 0, we have

∫R2n

ch(b♯n) = 1.

In particular (since b♯n generates Kc(R2n) ≅ Z, we get for each x ∈Kc(R2m):

∫R2m

ch(x) ∈ Z

The second statement is an integrality theorem. It is much deeper than Theorem9.3.6. Let us derive one consequence. Consider the identity

n

∏i=1

(t + xi) =n

∑k=0

tn−kσk(x1, . . . , xn).

Setting x = −ti and summing over i = 1, . . . , n yields

0 =n

∑k=0

(−1)n−ksn−k(x1, . . . , xn)σk(x1, . . . , xn)

or

0 =n

∑k=0

(−1)n−ksn−kck ∈ I(U(n))

which gives a recursive formula for the computation of sn. Now we consider acomplex vector bundle W → S2n. We can write W = V ⊕ Ck, V a bundle of rankn (bundles of higher rank have nowhere vanishing cross-sections and therefore splitoff a trivial line bundle). Since s0(V ) = n and since ck(V ) = 0 ∈ H2k(S2n) = 0 for0 < k < n, the above formula reads

0 = (−1)nsn(V )c0(V ) + s0(V )cn(V ) = (−1)nn!chn(V ) + ncn(V )


or

(−1)n−1 1

(n − 1)!cn(V ) = chn(V ).

Therefore, we get

1

(n − 1)! ∫S2ncn(V ) = (−1)n−1 ∫

S2nchn(V ) ∈ Z.

This is highly remarkable: it means that ∫S2n cn(V ) not only lies in Z, but in(n − 1)!Z (this is a special feature of the sphere: for example, on CPn, we get

∫CPn cn(H⊕n) = 1 by a (now) well-known calculation). We have proven

Theorem 9.7.8. For each vector bundle V → S2n, ∫S2n cn(V ) is a multiple of(n − 1)! (and for each such multiple there exists a vector bundle on the sphere withthis Chern number).

Corollary 9.7.9. For n ≥ 4, S2n does not have the structure of a complex manifold.

Proof. Assume that TS2n is a complex vector bundle. Then

2 = ∫S2n

e(TS2n) = ∫S2n

cn(TS2n)

and therefore 2 divides (n − 1)!, which is only possible if n ≤ 3.

We have seen that the only spheres which could have complex structures are S0,S2, S6. It is known that TS6 does have the structure of a complex vector bundle(or, equivalently, that S6 has an almost complex structure). The question whetherthere is an almost complex structure S6 that comes from a complex structure iswide open (as of April 2014) and probably a very difficult question.

The role of the Chern character for the index theorem is foreshadowed by areformulation of the Riemann-Roch theorem. Let X be a Riemann surface of genusg and V → X be a holomorphic vector bundle of rank n. By the definition of theChern character, ch1 = c1 and so we get ch(V ) = r+ c1(V ). We can reformulate theRiemann-Roch theorem as

ind(∂V ) = ∫X(1 + 1

2c1(TX))ch(V )

(the first term will be written in a more compact form below). Let X be anyspace. The Chern character induces a homomorphism

ch ∶K−n(X) =K0(SnX)→H2∗(SnX).The suspension SX is covered by two copies of the cone CX, and the intersection

is X. Since the cone is contractible, we get from the Mayer-Vietoris sequence anisomorphism H∗(X) ≅ H∗+1(SX) and hence by iteration H∗(X) ≅ H∗+n(SnX).There is a sign choice in the Mayer-Vietoris sequence, and we pick signs as follows.Let a ∈H1(I, ∂I) be the generator represented by the affine singular simplex ∆1 →I = [−1,1] with e1 ↦ −1 and e0 ↦ 1. Let u ∈ H1(I, ∂I) be such that ⟨u, a⟩ = +1.Taking cross products with u×n yields a map

ΣnH∗(X)→H∗+n(X × In,X × ∂In) ≅H∗+1(SnX),

166 JOHANNES EBERT

the last isomorphism holds since X × ∂ ± 1→X × I has the homotopy extensionproperty). It is easy to check that this map is an isomorphism, and we use it as areplacement for the suspension isomorphism. Therefore, we obtain homomorphisms

(9.7.10) ch ∶K0(X)⊗R→Hev(X;R); ch ∶K−1(X)⊗R→Hodd(X;R).

Theorem 9.7.11. For each finite complex X, the Chern character maps 9.7.10are isomorphisms. If M is a manifold which is the interior of a manifold W withboundary, then ch ∶Kc(M)⊗R→Hev

c (M ;R) and ch ∶Kc(M×R)⊗R→Hoddc (M ;R)

are isomorphisms.

Proof. We first prove the result in the CW case, by induction on the number ofcells. The manifold statement will be derived later. If X = ∅, the statement istrivial. Let us assume that Y =X ∪Sn−1 Dn is obtained by attaching a single n-cell.We write p(n) ∈ 0,1 for the parity of n. Assume that

K−n(X)⊗R→H2∗+p(n)(X,R)

is an isomorphism for n = 0,1. We first claim that for all n ≥ 0, K−n(X)⊗R →H∗+p(n)(X;R) is an isomorphism. This follows from Bott periodicity, since thediagram

K−n(X)β //

ch

K−n−2(X)

ch

H∗+p(n)(X;R) Σ2// H∗+p(n)+2

commutes. This is because the Chern character, applied to the Bott class, isu ∈ H2(I2, ∂I2), by the computation 9.7.7. Now we use the long exact sequencesin K-theory and in cohomology od the pairs (Y,X) and the 5-lemma. However,there is a slight problem: we have to show that the Chern character commuteswith the boundary homomorphisms. This is indeed the case (up to sign). Insteadof this cumbersome checking, one can argue in a different way. Namely, we workwith CW complexes. For a cellular inclusion A ⊂ B, the quotient map B → B/Ainduces an isomorphism H∗(B/A) → H∗(B,A). We define the functor H(X) ∶=Hev(X;R). On CW pairs, the functor has the same formal properties as K-theory(the short exact sequence and that collapsing contractible subcomplexes inducesisomorphisms). But the long exact sequence was a formal consequence of theseproperties, and therefore we get a long exact sequence for the H-functor. With thisnew exact sequence, it becomes clear that the Chern character commutes with theboundary maps, since all maps in the sequence come from maps of spaces. Thisfinishes the proof when X is a finite CW complex.

To get the result for manifolds, one considers compact manifolds with boundary.It is known that a compact manifold has the homotopy type a finite CW complex.This is a quite hard result, and much less is required. Namely, each compactmanifold M is dominated by a finite simplicial complex X, in other words, there isa finite complex X and two maps j ∶M →X and r ∶X →M such that ri = id. Theproof can be found in the appendix to [34]. We get a diagram


K(X) ≅ //

j∗

H(X)

j∗

K(M) //

r∗

H(M)

r∗

K(X) ≅ // H(X)

and an easy diagram chase shows that K(M)→H(M) is an isomorphism. WhenM is the interior of W , use the long exact sequences for the pair (W,∂W ).

9.8. Chern character and Thom isomorphisms. In the next section, we willderive the cohomological version of the index theorem from the K-theoretic one.We will face the following problem. Let V π ∶→X be a complex vector bundle over acompact space. There are Thom isomorphisms K0(X) → K0

c (V ) and H∗(X;R) ≅H+ − c(V ;R). Recall that these are given by Thom classes t = tV ∈ K0

c (V ) andτ ∈H2n(V ) and by

thK(x) = π∗t ⋅ t; thH(ω) = π∗ω ⋅ τ.Hence

(thH)−1ch(thK(x)) = (thH)−1(ch(π∗x⋅t) = (thH)−1(ch(π∗x)⋅ch(t)) = ch(x)⋅(thH)−1(ch(t));the last equation holds since the cohomological Thom isomorphism is a homo-

morphism of H∗(X)-left-modules.

Definition 9.8.1. Let V →X be a complex vector bundle. Define µ(V ) ∈H∗(X;R)by µ(V ) = (thH)−1(ch(t)).

The class µ is defined so that the diagram

K(X) thK //

x↦ch(x)µ(V )

Kc(V )

ch

H∗(X) thH // H∗

c (V )commutes. Since the Thom classes and the Chern character are natural, it is clear

that µ(V ) is a characteristic class (µ(f∗V ) = f∗µ(V )). Moreover, if V,W → X,Yare two vector bundles, we get (since the Thom classes are multiplicative) that

µ(V ⊕W ) = µ(V )µ(W ).By Cartans theorem, µ (for n-dimensional vector bundles) is given by an element

µn ∈ I(U(n)) = R[[x1, . . . , xn]]. By the multiplicativity, we get that

µn(x1, . . . , xn) =n

∏i=1

µ1(xi).

Let us determine the element µ1. For this, we have to understand the class µ fora line bundle, and by the classification of vector bundles, it is enough to study thecase of the tautological line bundle on CPn. Let V be a finite-dimensional C-vector

168 JOHANNES EBERT

space and consider the dual bundle π ∶HV → PV . There is a map j ∶HV → P(V ⊕C)given by sending (`, f) ∈ HV to the graph Γf ⊂ V ⊕ C of the map f ∶ ` → C. Themap j is an embedding, and it is a diffeomorphism onto the set P(V ⊕C)∖0⊕C.The bundle HV ⊕C has the cross section s, whose value at m ∈ P(V ⊕ C) is therestriction of the linear form given by the projection p ∶ V ⊕ C → C. The zero setof s is PV , and s defines a K-cycle C → HV ⊕C, (m,z) ↦ zs(m). This K-cycle

represents HV ⊕C − 1 ∈ K0(P(V ⊕C)), and we claim that j pulls back this K-cycleto the conjugate of the Thom clas of HV . This is easy to check. Let us summarize.

Lemma 9.8.2. Under the isomorphism Kc(HV ) ≅ K0(H+V ) = K0(P(V ⊕C)), the

thom class t −HV corresponds to 1 − [HV ⊕C] = 1 − [LV ⊕C].

Write xV ∶= c1(HV ) ∈H2(P(V )) It is much easier to show that the cohomologicalThom class τHV is xV ⊕C under these isomorphisms. Now the computation of µ(HV )is easily finished:

thH(µ(HV )) = ch(1 − [LV ⊕C]) = 1 − e−xV ⊕C ⇒ µ(HV ) = 1 − exVxV

.

This formula is to be understood as follows: the power series 1−e−xx

∈ R[[x]] =I(U(1)) is µ1. Of interest to us is less the class µ(V ), but its multiplicative inversemu(V )−1 ∈H∗(X;R). Recall the Taylor expansion

x

1 − e−x= 1 + x

2+∑k≥1

(−1)k+1 Bk(2k)!

x2k;

Bk ∈ Q is the Bernoulli number ; the first values are

B1 =1

6; B2 =

1

30; B3 =

1

42; B4 =

1

30. . .

For more information on Bernoulli numbers, see Appendix B of [52]; note thatthere are several customary conventions for Bernoulli numbers. We define the Toddclass Td ∈ I(U(n)) by

Td(x1, . . . , xn) =n

∏j=1

xj

1 − e−xj.

9.9. The cohomological version of the index theorem. Now we consider aclosed manifold M ⊂ Rn. Let U be a tubular neighborhood of M . Recall that thetopological index homomorphism was the composition

Kc(TM)thνM⊗C→ Kc(TU)→Kc(R2n)

Bott≅ Z.

Now we use the Chern character and get

Kc(TM)

ch( )µ(νC)

thK // Kc(TU) //

ch

Kc(R2n)

ch

β−n

##H∗c (TM) thH // H∗

c (TU) // H∗c (R2n) ∫ // R.

This diagram is commutative; the first square by the definition of the class µ, thesecond one because the maps in K-theory and cohomology are induced by maps ofspaces, and the triangle commutes by Theorem 9.7.7. We obtain, for x ∈Kc(TM):


top − ind(x) = ∫TU

(ch(x)µ(νM⊗C))τνM⊗C = ∫TM

(ch(x)µ(νM ⊗C));

the last equation holds by the definition of the cohomological Thom class. SinceνM ⊕ TM is trivial, we get that

µ(νM ⊗C)) = Td(TM ⊗C)).Now let π ∶ TM →M , and we get that

top − ind(x) = ∫TM

(ch(x)π∗Td(TM ⊗C)).

If M is oriented, we can reduce the integral to an integral over M :

∫TM

(ch(x)π∗Td(TM ⊗C)) = ε∫M

th−1H (ch(x))Td(TM ⊗C).

The symbol ε denotes a sign; it comes from the fact that if M is oriented, thereare two different orientations on TM . One comes from viewing TTM as a complexvector bundle; TTM ∣M = TM ⊕ TM ; the first summand is the tangent direction,

the second one the normal direction. The map j = ( −11

) is a complex struc-

ture on TTM , and by the orientation convention for complex vector bundles weobtain an orientation of the manifold TM . The other orientation comes from theorientation convention for oriented bundles over oriented manifolds. These orien-tation conventions boil down to the two conventions for orienting a complex vectorspace. If (e1, . . . , en) is a C-basis, we can either declare (e1, ie1, e2, ie2, . . . , en, ien)or (e1, . . . , en, ie1, . . . , en) to be positively oriented. Both conventions differ by thesign

ε = (−1)∑n−1i=1 i = (−1)n(n−1)/2.

Thus we have proven that the K-theoretic index theorem is equivalent to thefollowing formula.

Theorem 9.9.1. Let D be an elliptic differential operator on the closed orientedn-manifold. Then

ind(D) = top − ind(D) = (−1)n(n−1)/2 ∫M

th−1H (ch(σ(D)))Td(TM ⊗C).

Definition 9.9.2. Let D be an elliptic operator on M . The index class of D isI(D) = (−1)n(n−1)/2th−1

H (ch(σ(D)))Td(TM ⊗C).

We will restrict to even-dimensional manifolds from now on. This tames the signquestions considerably. The true reasons to do that is

Theorem 9.9.3. Let M be a closed odd-dimensional manifold. Then the topologicalindex of any elliptic differential operator on M is zero.

Proof. Let T ∶ TM → TM be the antipodal map. The symbol of D has the symme-try property that T ∗σ = (−1)rσ, where r is the order of D. In complex K-theory,the sign does not play a role, and so we get that T ∗σ(D) = σ(D). Since the degreeof the antipodal map is (−1)n = −1, we get that

170 JOHANNES EBERT

∫TM

(ch(x)π∗Td(TM⊗C)) = −∫TM

(T ∗ch(x)π∗Td(TM⊗C)) = −∫TM

(ch(x)π∗Td(TM⊗C)).

9.10. universal symbols. It remains to compute th−1H (ch(σ(D)) for operators

of interest. For concrete calculations with the index formula, we only consideroperators whose symbols are ”universal” in a sense that we now make precise.

Let G be a compact Lie group and ρ ∶ G → SO(2m) be a homomorphism. Weassume that V0, V1 are complex G-representations, and σ ∶ R2m → Hom(V0, V1) is aG-equivariant map, homogeneous and polynomial of degree r (the case r = 1 is themost interesing) and such that σ(x) is an isomorphism for all x ≠ 0. We explicitlysay that Hom(V0, V1) is the space of linear maps, without G-equivariance. We callthese data

c = (V0, V1, σ).Let P →X be a G-principal bundle on a compact space. We can form the vector

bundle E = P ×G R2m; and we obtain a compactly supported K-cycle from V0, V1

and σ on E, namely

c(P ) = [P ×G (R2n × V0), P ×G (R2n × V0), σ(P )] ∈Kc(E),where σ(P ) sends [p, x, v]↦ [p, x, σ(x)v]. We define

(9.10.1) λc(P ) ∶= (−1)nth−1H (ch(c(P ))) ∈H∗(M).

The assignment P ↦ λc(P ) ∈ H∗(X) is a characteristic classes of G-principalbundles on closed manifolds and hence, by Cartan’s theorem, given by an elementλc ∈ I(G). Let M be a closed oriented 2n-manifold, P → M a principal bundleand let there be chosen an (orientation-preserving) isomorphism P ×G R2m ≅ TM .In other words, the tangent bundle TM comes with a reduction of the structuregroup to G. In this case, the map σ(P ) ∶ P ×G (R2n × V0)→ P ×G (R2n × V1) is thesymbol of an order r elliptic operator Dc, with symbol class is c(P ). Of course,the operator Dc is only defined up to lower order operators, which do not affectthe index. In this case, the index theorem reads

ind(Dc) = ∫Mλc(P )Td(TM ⊗C).

We want to compute the element λc in terms of representation-theoretic data.We need some very basic facts from the representation theory of compact Lie groups.Let G be a compact Lie group and let Rep(G) be the set of isomorphism classes offinite-dimensional complex representations of G, which is a commutative semiring.The representation ring R(G) of G is by definition the Grothendieck group ofRep(G). For each representation G V , there is an invariant hermitian metric(use the Haar integral), and so any V ∈ Rep(G) is given by a homomorphismφ ∶ G→ U(n), for some n. We define

ch(V ) ∶= φ∗ch ∈ I(G).Clearly, this defines a ring homomorphism ch ∶ R(G) → I(G). Moreover, if

P → M is a G-principal bundle, we get a homomorphism Rep(G) → Vect(M);


V ↦ P ×G V , which gives a ring homomorphism R(G) → K(M). The followingdiagram commutes:

R(G) ch //

I(G)

CW(P, )

K(M) ch // H∗(M ;R).Since each representation V is unitary, V can be written as a direct sum of

irreducible representations. If G is abelian, then each irreducible representation isone-dimensional (Schur’s lemma). Let U be the standard representation of T; andlet Uj be the representation of T(n) given by projection to the jth factor. Theirreducible representations of T(n) are U⊗r1

1 ⊗ . . . U⊗rnn .

We need two assumptions on ρ. We assume that there is a maximal torus T ⊂ Gsuch that ρ(T ) ⊂ D(n). This is not always satisfied (take a subgroup of SO(2m)which does not intersect D(n)), but can be adjusted by conjugating ρ in SO(2m)),but is merely a cosmetic assumption. The essential constraint is:

Assumption 9.10.2. The element ρ∗e ∈ Im(G) is nonzero.

To get a feeling for this assumption, note that ρ∗e ≠ 0 if and only if the restrictionof ρ∗(e) to T is nonzero (by Corollary 9.4.11). So to discuss the assumption, it isenough to consider the case G = T .

Proposition 9.10.3. Let T be a torus and ρ ∶ T → D(n) be a representation. Thefollowing conditions are equivalent.

(1) ρ∗(e) ≠ 0 ∈ I(T ).(2) The action of T on S2n−1 induced by ρ does not have a fixed point.

If ρ is surjective, these conditions are satisfied.

Proof. Let ρi be the ith component of ρ. If the T -action on S2n−1 has a fixed point,then the decomposition of ρ into irreducibles must contain a trivial summand. Inother words, there is a j such that ρj ≡ 1. Since ρ∗(e) = ∏n

j=1 ρ∗j yj , this implies

that ρ∗(e) = 0. If ρ∗(e) = 0, then one of the ρ∗j (yj) must be zero (since I(T ) has nozero-divisors). This means that ρj is constant, and hence that T has a fixed point.

If ρ is surjective, then ρ∗ ∶ I(T(n))→ I(T ) is injective, and so ρ∗(e) ≠ 0.

After these preliminaries, we can return to the computation of the class λc ∈I(G). The ring I(G), being a subring of a power series algebra, has no zero-divisors, and therefore no information is lost if we compute λρ∗(e) instead.

Proposition 9.10.4. λcρ∗(e) = (−1)nch(V0 − V1) ∈ I(G).

Proof. It is enough to check this relation on a closed manifold, after applying theChern-Weil construction. For any oriented vector bundle π ∶ V →M over a closedmanifold, with zero section ι, the relation

(9.10.5) th−1H (x)e(V ) = ι∗x ∈H∗(M)

holds for each x ∈H∗c (V ). To verify this relation, write x = thH(y) for y ∈H∗(M)

by the Thom isomorphism theorem and use the definition of the Euler class.

172 JOHANNES EBERT

Again, because I(G) has no zero-divisors, we can divide the above relation by thenonzero element ρ∗(e). The resulting equation a priori only holds in the localization

I(G)[ 1ρ∗(e) ], but we know in advance that λc ∈ I(G) and since there are no zero-

divisors, the map I(G)→ I(G)[ 1ρ∗(e) ] is injective.

Corollary 9.10.6. If the assumption 9.10.2 holds, then

λc = (−1)n ch(V0 − V1)ρ∗(e)

.

Let zj ∶= ρ∗yj ∈ I(T ). Let ρC ∶ G→ U(2n) be the complexification. The universalindex class is

Ic ∶= λcρ∗CTd.

Inserting the formula for the Todd class and 9.10.6, we compute

(9.10.7)

Ic = (−1)n ch(V0 − V1)∏nj=1 yj

n

∏j=1

yj

1 − e−yj−yj

1 − eyj= ch(V0 − V1)

n

∏j=1

xj

(1 − e−xj)(1 − exj)

This is the formula we were aiming at all the time.Before we go to really concrete examples, let us introduce a sometimes useful

construction with differential operators. Let M be a closed manifold and D ∶Γ(M,V0)→ Γ(M ;V1) be an elliptic operator, with symbol symb(D) ∶ π∗V0 → π∗V1

and index class I(D) ∈H∗(M). Let E →M be another complex vector bundle. Wetensorize the symbol with idE and obtain an elliptic symbol π∗V0 ⊗E → π∗V1 ⊗E.By the symbol DE , we denote any elliptic operator with this symbol, and call DE

the twisting of D with E. Of course, DE is only defined on the symbolic level(later, when D is a Dirac operator, we will obtain a canonical operator DE . Anexample for a concrete twisting appears when we consider the Dolbeault complexon a complex manifold and a holomorphic vector bundle E →M . In this case, theDolbeault operator on E is the twisting of D by E. At the moment, we only notethe relation of the index classes:

I(DE) = I(E)ch(E),which after all the work we did is quite obvious.

9.11. The classical index theorems. We now specialize our calculations to theclassical operators: the Euler characteristic operator, the signature operator andthe Dolbeault operator.

9.11.1. The Euler characteristic. In this case G = SO(2n), ρ = id, V0 = Λev2n =ΛevR2n, V1 = Λodd2n = ΛevR2n. We have to compute ch(Λev)−ch(Λodd) ∈ I(SO(2n)).This is easy, because of the formula

(Λev2n −Λodd2n ) = (Λev2 −Λodd2 ) ⊠ . . . ⊠ (Λev2 −Λodd

2 )(n times), which takes place in the representation ring R(D(n)) of the maximal

torus. The Chern character of Λev2 −Λodd2 is easy to compute; the result is 2−ey−e−y =−(ey/2 − e−y/2)2. We compute


λ =n

∏j=1

−(2 − eyj − e−yj)yj

and

I =m

∏i=1

(2 − eyi − e−yi)m

∏i=1

yi(1 − e−yi)(1 − eyi)

=m

∏i=1

yi.

Good news! The universal index class is the Euler class, and this confirms whatwe already know. We summarize

Theorem 9.11.1. The cohomological version and hence the K-theoretic version ofthe index theorem holds for the Euler characteristic operator.

This should give credibility that the cohomological computations are correct, inparticular, it gives a sanity check for the signs. On the other hand, we use theGauss-Bonnet-Chern theorem in the proof of the index formula. Note that bothpurposes certainly overload this computation. We will have other opportunities tocheck the signs.

Let us look at the formula for λ a bit closer. Recall that λ measures the imageof the Chern character of the symbol under the inverse Thom isomorphism. Letus consider the symbol class, restricted to a point in M . This is an element ofKc(TxM) and hence some multiple of the Bott class (by periodicity). To determinewhich multiple, we have to calculate the degree 0 term of λ. But

−(2 − ey − e−y)y

= −2 − 1 − y − y2/2 − 1 − +y − y2/2 + . . .y

≡ 0 (mod y),

and so this multiple is zero. In other words, the symbol of the Euler characteristicoperator, restricted to each fibre, is zero. There is of course a direct way of seeingthis. Also, note that the index class is homogeneous (i.e. it has no terms exceptin degree 2n). This is by no means a general phenomenon, as we will see. Oneconsequence is

Corollary 9.11.2. (Conditional) Let M be an oriented 2n-manifold and E a com-plex vector bundle. Then the index of the Euler operator, twisted by E, is the sameas χ(M) rank(E).

This shows that the Euler characteristic operator really does not contain anyinformation, besides the Euler number of the manifold. If this were true for alloperators, then there would be no index theorem. The Euler operator is quiteuninteresting from the general viewpoint of index theory (but we will use the Gauss-Bonnet theorem).

9.11.2. The Hirzebruch signature theorem. In this case G = SO(2n), ρ = id. Recallthat we defined an involution τ = τn on (Λ∗R2n)⊗C and that the signature operatoris the graded differential operator (D = d+d∗) (D; τ). The involution τ is on p-forms

given by ip(p−1)+n⋆.Let Λ±

n be the ±-eigenspace of τ . These are SO(2n)-representations. We haveto compute the element

ch(Λ+n −Λ−

n) ∈ I(SO(2n)).

174 JOHANNES EBERT

We can restrict to the maximal torus D(n) for this purpose. Let ω ∈ Λ∗R2n andη ∈ Λ2m, with cross-product ω×η ∈ Λ∗R2m+2n. It is easy, but quite tedious to checkthat

τn+m(ω × η) = τn(ω) × τm(η).

This formula implies that the restriction of Λ+n −Λ−

n ∈ R(SO(2n)) to D(n) is thesame as

(Λ+2 −Λ−

2) ⊠ . . . ⊠ (Λ+2 −Λ−

2)

(n times). Therefore, it is enough to compute Λ+2 −Λ−

2 . This can be done directly,by computing τ on the standard basis. The results are that u0, u1 are a basis forΛ+

2 and v0, v1 are a basis for Λ−2 , with

u0 = 1 + ie1 ∧ e2; v0 = 1 − ie1 ∧ e2; u1 = e1 + ie2; u2 = e1 − ie2.

The action of the element z = a + ib ≙ (a −bb a

) in T ≅ D(1) is given by

z∗u0 = u0; z∗v0 = v0; z∗u1 = zu1; z∗u2 = zu2.

Thus Λ+2 = 1 +U1 and Λ−

2 = 1 −U−1 ∈ R(T). Therefore

ch(Λ+2 −Λ−

2) = ey − e−y ∈ I(SO(2))

and

ch(Λ+n −Λ−

n) =n

∏j=1

(eyj − e−yj).

Therefore

λ =n

∏j=1

eyj − e−yjyj

and

I =n

∏j=1

yj(eyj − e−yj)(1 − e−yj)(1 − eyj)

.

But

y(ey − e−y)(1 − e−y)(1 − ey)

[Check signs carefully]

Theorem 9.11.3. (The Hirzebruch signature theorem)

The signature theorem is much more profound.

Corollary 9.11.4. Let f ∶M → N be an d-sheeted cover of oriented 4k-manifolds.Then sign(M) = d sign(N).


This is immediate; by the signature theorem and because ∫M f∗ω = d ∫N ω fordifferential forms. The corresponding formula for the Euler characteristic is quiteclear, so your initial reaction might be that the corollary might follow from Poincareduality. This is not the case - Wall constructed a CW complex which satisfiesPoincare duality, but not the multiplicativity of the signature. More important forus is at the moment that the order 0 term of the class λ is ±2n. As a consequence,the symbol of the signature operator, when restricted to a fibre, is ±2n times theBott class (life would be greatly simplified if the factor were 1, but there is anobstruction against that). In particular, we note

Corollary 9.11.5. On any oriented (nonempty) manifold M of even dimension,the symbol class of the signature operator σ(D) ∈Kc(TM) is nontrivial.

One can check that the signature theorem is true for CPn. Consider the sphereS2n. The L-class is a polynomial in the Pontrjagin classes. As the sphere is stablyparallelizable, it follows that the L-class is zero. Also, since Hn(S2n) = 0, thesignature of the sphere is zero. Therefore we have checked

Theorem 9.11.6. The index theorem (both versions) is true for the signatureoperator on the sphere.

Because the symbol is nonzero, this has some content (!). In fact, Theorem9.11.1 and 9.11.6 are the two base cases for the index theorem that enter our proofof the index theorem.

The Hirzebruch signature theorem was proven one decade before the index theo-rem. Let us sketch Hirzebruch’s proof. First, one proves that the theorem holds forCPn. This is not entirely trivial, but a nifty calculation using the computation ofthe characteristic classes of CPn. Consider two cobordant manifolds M0 and M1.It is quite easy, using P (V ⊕R) = p(V ), to prove that the L-genera of M0 and M1

coincide (this is definitely false for the Euler class). A fundamental property of thesignature is that it is cobordism invariant as well (not trivial; if M = ∂W , then thenimage of H∗(W ) → H∗(M) is a Lagragian subspace). Let Ωn be the cobordismgroup of n-dimensional oriented manifolds and Ω∗ = ⊕nΩn. It is easy to see thatM ↦ L(M) gives a ring homomorphism Ω∗ → Q (a priori not to Z!). Moreover,sign(M ×N) = sign(M) sign(N) (Kunneth).

Thom computed Ω∗, at least Ω∗ ⊗Q, and the result is that it is a polynomialalgebra on the generators CP2,CP4,CP6 . . .. Therefore, the signature theorem istrue. Thoms proof was in two steps. The first is the reduction to a homotopytheoretic problem, using the famous Pontrjagin-Thom construction. Then he usedresults by Serre on rational homotopy theory and characteristic classes.

The first proof of the index theorem by Atiyah and Singer used the signaturetheorem. In a nutshell, the argument is as follows. By extending the class of dif-ferential operators to pseudodifferential operators, one proves that each class inK(TM) is the symbol class of some elliptic operator (false for differential oper-ators!). Moreover, the index only depends on the symbol class. Therefore, theanalytical index is given by a homomorphism K(TM)→ Z.

9.11.3. The Hirzebruch-Riemann-Roch theorem. Let M be a complex manifold ofdimension m. We have seen the Dolbeault complex. This is universal for the groupU(m) and the representation ρ ∶ U(m) → SO(2m). Recall that ρ∗yi = xi. Therepresentations Vi are

176 JOHANNES EBERT

V0 = ΛevCm; V1 = ΛoddCm

(note that Λ0,pV ∗ is the pth exterior power of the space of C-antilinear mapsV → C, and hence ΛpV ). Again, the virtual representation ΛevCm − ΛoddCm ismultiplicative, and for m = 1, we get ch(C − Cm) = 1 − ex. Altogether, the indexclass is

λ =m

∏i=1

(1 − exixi(1 − exi)(1 − e−xi)

=n

∏i=1

xi(1 − e−xi)

= Td(x1, . . . , xm).

Therefore, we get

ind(∂ + ∂∗) = ∫M

Td(TM).

When we twist the Dolbeault complex with a complex vector bundle, we obtain

Theorem 9.11.7. (The Hirzebruch-Riemann-Roch theorem) ind(∂V +∂∗V ) = ∫M ch(V )Td(TM).

We perform now two sanity checks. If dim(M) = 1, we should obtain theRiemann-Roch theorem. Indeed, Td(TM) = 1 + 1

2c1(TM) and ch(V ) = rank(V ) +

c1(V ). The other example is M = CPn and V = C. In this case, we had TCPn⊕C =H⊕(n+1), therefore

∫CPn

Td(TCPn) = ∫CPn

( x

1 − e−x)n+1

9.12. Literature. The standard reference for characteristic classes is [52], but thissource works in singular cohomology. For the general theory and the proof ofCartans theorem, I followed [23]. The treatment of the maximal torus theoremseems to go back to Adams [1]; some more details appear in [21] and [19], p. 388ff.The books by Adams and Brocker-tomDieck also contain important backgroundmaterial on compact Lie groups. The master of computations with characteristicclass was without doubt Friedrich Hirzebruch. You find many more examples ofcomputations in his collected works. Hirzebruch’s early work [37] initiated thedevelopment of the index theorem; in particular, he proved the signature theoremand (an important special case of) the Hirzebruch-Riemann-Roch theorem in thatbook. The translation of the K-theoretic formulation of the index theorem into thecohomological one was done by Atiyah and Singer in [9].


10. Dirac operators and proof of the index theorem: the easy part

We will only prove the index theorem for a special class of operators, the Diracoperators. This has a couple of reasons.

(1) All classical operators are Dirac operators.(2) There is an intimate relationship between the linear algebra that underlies

the Dirac operators and K-theory.(3) We use the special algebraic structure of Dirac operators in an essential way

during the proof of the index theorem. For general operators, we do nothave such a sensible connection between the operator and K-theory. In fact,I do not know how to prove the index theorem for differential operators,without introducing a more general class of operators, the pseudodifferentialoperators.

(4) If one knows the calculus of pseudodifferential operators, the most generalversion of the index formula can be derived from the index formula forDirac operators by a fairly simple argument.

(5) The argument that we give is just marvellous!

10.1. The definition. Before we give the definition, let us make a convention.From now on, all real vector bundles will be Riemann vector bundles and all mani-folds will be Riemannian manifolds by default, and we will very often identify T ∗Mand TM using the musical isomorphisms. Furthermore, each vector bundle will beequipped with a bundle metric. In the beginning, we do not specify whether thebundle is real or complex. As usual, the real theory is more complicated, and wemake no attempt at developping the real theory. However, we try at least to sayexplicitly where we used the complex numbers.

Often, a vector bundle E → X (or a single vector space) will have a grading, bywhich we understand a decomposition E = E0 ⊕E1, and we require the decomposi-tion to be orthogonal. An equivalent and much more clever way to think about agrading is to consider the linear map ι ∶ E → E which has Ei as eigenbundle to theeigenvalue (−1)i. This map is an involution ι2 = id, and that E0E1 is expressedby saying that ι is self-adjoint. A map f ∶ E → F between graded vector bundles iscalled even if fι = ιf and odd if fι = −ιf . Likewise conventions apply to differentialoperators.

The tensor product of two graded bundles (E, ι) and (F, τ) is (E ⊗F, ι⊗ τ). Interms of the decomposition, this is the same as saying that

(E ⊗ F )0 = E0 ⊗ F0 ⊕E1 ⊗ F1; (E ⊗ F )1 = E1 ⊗ F0 ⊕E0 ⊗ F1.

If we decompose E = E0 ⊕E1 and write the involution as ι = (1−1

), then an

even or odd map is of the form

(ab) ; or ( a

b) .

A graded algebra is a graded vector space (A, ι), equipped with a bilinear mapA⊗A→ A. This can be phrased by saying that AiAj ⊂ Ai+j .

Definition 10.1.1. Let M be a Riemannian manifold, E → M a smooth Rie-mannian or Hermitian vector bundle. A Dirac operator is a first order differential

178 JOHANNES EBERT

operator D ∶ Γ(M ;E)→ Γ(M ;E) such that D is formally self-adjoint and such thatthe symbol smb1(D) satisfies

smb1(D)(ξ)2 = ∥ξ∥2;

in the sense that the left-hand side is the scalar operator given by multiplicationby the square of the norm.

A graded Dirac operator is a triple (E,D, ι), where E is a Riemannian/Hermitianvector bundle, ι a self-adjoint involutive bundle automorphism, and D a Diracoperator on E such that Dι = −ιD.

Note that a Dirac operator is automatically elliptic. While an ungraded Diracoperator does not have a nonzero index, a graded one does. Indeed, the involutionι splits E as a direct (orthogonal) sum E = E + ⊕E− and the condition Dι = −ιDenforces that D is of the form

( D∗0

D0) .

Thus, on any closed manifold M , a graded Dirac operator has an index

ind(D, ι) ∶= ind(D0).The opposite grading is just the grading −ι, note that ind(D, ι) = − ind(D,−ι).

Recall that the symbol of an order one operator D is given by

smb1(D)(df) = i[D,f].When dealing with Dirac operator, it is convenient4 to omit the factor of i. Let

c(df) = [D,f], which gives a linear map c ∶ TxM → End(Ex), and the condition fora Dirac operator means that

(10.1.2) c(ξ)2 = −∥ξ∥2.

That D is self-adjoint implies that

(10.1.3) c(ξ)∗ = −c(ξ).

10.2. The classical operators are Dirac operators. A large portion of thischapter will surround the algebra of the two identities 10.1.2 and 10.1.3, but beforewe begin with this investigation, we discuss basic examples. Let V be a euclidean,finite-dimensional vector space and Λ∗V its exterior algebra. For each v ∈ V , thereis the map εv ∶ Λ∗V → Λ∗V that takes ω ↦ v∧ω, and the insertion operator ιv. Theexterior algebra has a canonical inner product, and it is easy to see that with respectto this inner product, the operators ιv and εv are mutually adjoint. Moreover

(εv − ιv)2 = −∥v∥2.

Consider the de Rham complex of a manifold M . The symbol of the exteriorderivative d is given by

smbd(ξ) = iεξ.

4When treating the theory of real operators, one needs to be extremely careful here!


Consider D = d + d∗. Then

c(ξ) = εξ − ιξand hence D is a Dirac operator. On any Riemann manifold M , we have the

even/odd grading of the exterior algebra (I = (−1)p), and (D; I) is a graded Diracoperator, the Euler characteristic operator. If M is oriented and even-dimensional,there is another grading defined by the Hodge star. Namely, τ = ip(p−1)+m⋆ and weobtain the signature operator as a graded Dirac operator.

In a similar way, we can consider a complex manifold M with hermitian met-ric, and a holomorphic vector bundle V → M . There was the Dolbeault complexA0,∗(M ;V ) and the operator ∂ on it. One can likewise prove that 1√

2(∂ + ∂∗) is a

Dirac operator.

10.3. Clifford bundles. Let us analyze the underlying bundle data of a Diracoperator. Let K one of the fields R or C; when it gets serious, we consider only thecase K = C.

Definition 10.3.1. Let V →X be a Riemannian vector bundle. A Clifford bundleover V or Cl(V )-bundle is a pair (E, c), where E →X is a hermitian vector bundleand c ∶ V → End(E) is a linear bundle map such that

c(v)2 = −∥v∥; c(v)∗ = −c(v)holds for all vectors v ∈ V . A graded Clifford bundle over V is a triple (E, c, ι)where (E, c) is a Clifford bundle and ι ∶ E → E is a self-adjoint involution such that

c(v)ι = −ιc(v)holds for all vectors v ∈ V .

By polarization, one sees that the condition c(v)2 = −∥v∥2 is equivalent to

c(v)c(w) + c(w)c(v) = −2⟨v,w⟩.We use the shorthand notation (v ∈ Vx, e ∈ Ex)

v ⋅ e ∶= c(v)(e).

Remark 10.3.2. When I first attempted to learn the theory, I thought that thegrading seem is an unnecessary complication, and I was unwilling to take it serious.Only later I understood that the grading is an essential part of the algebra andmust not be ignored. In fact, it greatly simplifies everything. In fact, we willalmost exclusively consider graded Clifford modules, the ungraded ones make ashort auxiliary appearance.

Proposition 10.3.3. Let M be a Riemann manifold and (E; ι) a graded hermitianvector bundle on M . A graded Dirac operator D ∶ Γ(M ;E)→ Γ(M ;E) induces thestructure of a graded Cl(TM)-module on E, via the formula

c(v) ∶= −ismb1(D)(v).Vice versa, if (E, ι, c) is a graded Cl(TM)-bundle, then there exists a graded Diracoperator D on E that induces the given Cl(TM)-module structure on E. The index

180 JOHANNES EBERT

of D only depends on the Cl(TM)-module structure. The same statements hold forungraded objects.

Proof. It is clear that the formula for c(v) defines a Cl(TM)-structure. We haveproven in the previous term that there exists a differential operatorA with smb(A)(ξ) =ic(ξ). Let B = 1

2(A+A∗). The operator B is formally self-adjoint and has the same

symbol as A (since the symbol of A is self-adjoint), and so B is an ungraded Diracoperator. The operator D = 1

2(B − τBτ) is the desired graded Dirac operator. Two

graded Dirac operators have the same symbol and hence the same index.

The most interesting examples of Dirac operators are the geometric Dirac oper-ators, which bear a close relationship with the underlying Riemann metric on M .However, the immediate task we have in mind is the proof of the index theorem,and for this purpose these geometric operators are irrelevant, and we postpone thediscussion of them, and we turn to the discussion of the linear algebra that underliesthe Dirac operators.

10.4. Algebra. We defined Cl(V )-modules when V is a euclidean vector space (orbundle). It is conceptually helpful (and essential for the extension of the theory tothe real case) to consider also nondegenerate symmetric bilinear forms on V . Thebasic reference for Clifford algebras is the paper [6] by Atiyah, Bott and Shapiro.A more extensive exposition is in [48].

Definition 10.4.1. Let V be a R vector space with a symmetric bilinear form b.A Clifford algebra of (V, b) is a pair (Cl(V ), i), consisting of a unital associativeR-algebra Cl(V ) and a linear map i ∶ V → Cl(V ), such that i(v)i(w) + i(w)i(v) =−2b(v,w)1 and such that the following universal property holds. If g ∶ V → A isa linear map to an associative unital R-algebra such that g(v)g(w) + g(w)g(v) =−2b(v,w)1, then there is a unique algebra homomorphism h ∶ Cl(V ) → A withh i = g.

As usual, the Clifford algebra is, up to isomorphism, uniquely determined bythese properties. A construction is not difficult. Take a basis (vj)j∈I of V , letbjk = b(vj , vk), and take the free associative algebra generated by the basis. Thendivide out by the 2-sided ideal, generated by the elements

vjvk + vkvj + 2bjk.

It is easy to see that for the linear form b = 0, we get the exterior algebra.From now on, we confine ourselves to finite-dimensional, real vector spaces andnondegenerate bilinear forms.

It is also clear, from general nonsense, that V ↦ Cl(V ) is a functor from vectorspaces equipped with symmetric bilinear forms and maps preserving the forms toassociative algebras: let f ∶ (V, b) → (W,c) be an isometry (c(fv, fw) = b(v,w)).Consider the map i f ∶ V → Cl(W ). By the universal property, there is a uniquealgebra map f ′ ∶ Cl(V ) → Cl(W ) such that f ′ i = i f . In particular, the groupO(V ) of isometries of a vector space acts by algebra isomorphisms on Cl(V ), andit is easy to see that this action is continuous.

The Clifford algebra is graded: the map −i ∶ V → Cl(V ) satisfies the assumptionsof the universal property and induces an algebra isomorphism ι of Cl(V ) of order2. It is clear that ιi(v) = i(−v). It is true that the map i is always injective, andwe will drop it from the notation, and consider V as a subspace of Cl(V ).


Sylvester’s law of inertia states that each finite-dimensional real vector spaceV with a symmetric nondegenerate bilinear form b is isometric to the space Rp+q,equipped with the bilinear form

⟨x, y⟩ ∶= xT (1p1q

) y,

for uniquely determined p, q. We call this space Rp,q and denote its Cliffordalgebra by Clp,q. Observe that the Clifford algebra is generated by elementse1, . . . , ep, ε1, . . . , εq, subject to the relations

e2i = −1; ε2j = +1; eiej + ejei = 0; εiεj + εjεi = 0; eiεj + εjei = 0.

It follows that the dimension of Clp,q is 2p+q, and a basis is given by the elements

ei1⋯eil ⋅ εil+1⋯εik ∣i1 < . . . il; il+1 < ik; 0 ≤ l ≤ k ≤ p + q.We already mentioned that Clp,q is graded, and the even (odd) part is spanned

by those basis elements which contain an even (odd) number of factors. Let O(p, q)be the isometry group of Rp,q; it contains O(p) × O(q) as a maximal compactsubgroup. The action of O(p, q) on Clp,q is continuous.

Any vector space with symmetric bilinear form can be obtained in the followingway. A pseudo-euclidean vector space is a euclidean vector V space with innerproduct (, ), together with a self-adjoint involution τ on V (τ is literally a grading,but it plays a different role). Consider the bilinear form

bτ(v,w) ∶= (v, τw).which is symmetric because

(v, τw) − (w, τv) = (v, τw) − (τw, v) = 0.

We let Cl(V, τ) be the Clifford algebra of (V, bτ). There is another piece ofstructure on the Clifford algebra Cl(V, τ). Namely, consider the opposite algebraCl(V, τ)op and the map

V → Cl(V, τ)op; v ↦ −i(τ(v)).It satisfies

j(v)2 = i(τ(v))2 = −b(τv, τv) ∶= −(τv, v) = −(v, τv) =∶ −b(v, v).Hence it induces an isomorphism Cl(V )→ Cl(V )op, in other words an antiauto-

morphism a ↦ a∗ of Cl(V ). The norm of a is a∗a. In the Clifford algebra Clp,q,this isomorphism is given by e∗i = −ei and ε∗i = εi. It is a little exercise to verify that

(a∗)∗ = a.As you expected, we will now carry over the above definitions into a parameter-

ized setting. A pseudo-Riemannian vector bundle is a Riemannian vector bundleV → X with inner product (, ), together with a self-adjoint involution τ ∶ V → V .Let b(v,w) ∶= (v, τw) be the induced bilinear form.

Exercise 10.4.2. Interprete the above gadget at a fibre bundle with structural groupO(p) ×O(q) and fibre Rp,q.

182 JOHANNES EBERT

Let P →X be the underlying O(p)×O(q)-principal bundle. We form the algebrabundle

Cl(V, τ) ∶= P ×O(p)×O(q) Clp,q .

This algebra bundle carries a grading ι and an antiautomorphism ∗.

Definition 10.4.3. Let (V, τ) be as above. A (graded) Cl(V, τ)-module bundle isone of the following equivalent data.

(1) Triples (E, c, ι), where E →X is a hermitian vector bundle, ι a self-adjointinvolution and c ∶ Cl(V, τ)→ End(E) is a ∗-homomorphism of graded alge-bra bundles.

(2) Triples (E, c, ι), where (E, ι)→X is a graded hermitian vector bundle andc ∶ V → End(E) is a linear bundle map such that c(v) is odd, c(v)2 = (v, τv)and c(v)∗ = c(−τv).

Exercise 10.4.4. (1) Prove that both data are indeed equivalent.(2) You are probably confused by the fact that we used the symbol ι for two

different things. Figure out why this is not dangerous.

We will often use

c(v)e =∶ v ⋅ e.

Example 10.4.5. A simple minded example is when V → X is the zero bundle.In that case, a Clifford bundle is the same as a graded hermitian vector bundle.

10.5. The Atiyah-Bott-Shapiro construction. We now introduce two funda-mental constructions. The first is an external product.

Definition 10.5.1. Let (V, τ)→X and (W,σ)→ Y two pseudo-Riemannian vectorbundles and (E, c, ι), (F, d, η) be two Clifford modules. The exterior product E ♯Fis the Cl((V, τ) × (W,σ))-module over X × Y , given by the following data.

(1) The underlying hermitian vector bundle is E ⊠ F →X × Y .(2) The grading is ι⊗ η.(3) The Clifford action e ∶ V ×W → End(E ⊠ F ) is given by the formula

e(v,w) ∶= c(v)⊗ 1 + ι⊗ d(w).

It is quickly checked that this is indeed a Clifford module, the relevant compu-tation (omitting the symbols c and d) is

(v ⊗ 1 + ι⊗w)2 = v2 ⊗ 1 + vι⊗w + ιv ⊗w + ι2 ⊗w2 = v2 ⊗ 1 + 1⊗w2

which also clarifies the introduction of ι in the formula for e(v,w). The con-struction has many important special cases, and we will discuss the examples whenthey show up. The next construction lies at the very heart of the matter. As thename suggests, it was introduced in the paper [6].

Definition 10.5.2. Let V,W →X be two Riemannian vector bundles over a com-pact space and consider the pseudo-Riemannian vector bundle π ∶ (V,+)⊕(W,−)→X. Let (E, ι, c) be a graded Cl((V,+) ⊕ (W,−))-module. Let Ei be the (−1)i-eigenbundle of ι and let


γ ∶ π∗E0 → π∗E1

be the map which is given by ((v,w) ∈ (V ⊕W )x)

γv,w = c(v) + ic(w) ∶ π∗E0 = (E0)x → (E1)x = π∗E1.

Then (π∗E0, π∗E1, γ) is a compactly supported K-cycle on V ⊕W . We denote

abs(E) ∶= [π∗E0, π∗E1, γ] ∈Kc(V ⊕W ).

This K-theory class is called the Atiyah-Bott-Shapiro class of E.

Proof that this is indeed a K-cycle. We have to show that if (v,w) ≠ 0, then γv,wis an isomorphism. But (note that the same formula defines a map π∗E1 → π∗E0

so that we can talk about γ2v,w)

γ2v,w = (c(v) + ic(w))2 = c(v)2 + ic(v)c(w) + ic(w)c(v) − c(w)2 = c(v)2 − c(w)2 < 0

if (v,w) ≠ 0 (note that the factor of i is indispensable). So γv,w is invertible.

Example 10.5.3. Let M be a Riemann manifold E → M be a graded Cl(TM)-bundle and D be a graded Dirac operator on E. Then the Atiyah-Bott-Shapiroconstruction on E yields the symbol class of D:

abs(E) = σ(D).

Example 10.5.4. Let S1,1 be the following Cl1,1-module (here X is a point). Thevector space is C⊕C, with grading

ι = (1−1

)

and Clifford action given by

e = ( −11

) ; ε = ( 11

) .

We compute abs(S1,1). The bundles are R2 ×C. Over (x, y) ∈ R2, the map γ ismultiplication by x + iy. Hence we get the Bott element, or rather its negative

abs(S1,1) = −b ∈Kc(R2).

Exercise 10.5.5. Let V → X be a complex vector bundle, with compatible Rie-mannian metric. Find a Cl(V )-module (here V is viewed as a real vector bundle!!)E, such that abs(E) = tV , the K-theory Thom class.

Theorem 10.5.6. The Atiyah-Bott-Shapiro construction is compatible with prod-ucts in K-theory. More precisely, if (V, τ) → X and (W,σ) → Y are two pseudo-Riemannian vector bundles and (E, c, ι), (F, d, η) are two Clifford modules, then

abs(E ♯F ) = abs(E) ♯abs(F ) ∈Kc(V ×W ).The ♯ on the left-hand side is the product of Clifford modules, and on the right

hand side, it denotes the product in K-theory.

The proof is a straightforward exercise.

184 JOHANNES EBERT

Definition 10.5.7. Let V →X be a pseudo-Riemannian vector bundle and (E, ι, c)be a Cl(V )-module. We say that E is inessential if it extends to a Cl(V ⊕ (R,+))-module. In other words, we require the existence of a map e ∶ E → E that satisfies(for all v ∈ V )

e∗ = −e; e2 = −1; ec(v) = −c(v)e; eι = −ιv.

Lemma 10.5.8. Let (V,+) ⊕ (W,−) → X be a pseudo-Riemannian vector bundleand let (E, ι, c) be a Cl((V,+)⊕ (W,−))-module. If E is inessential, then abs(E) =0 ∈Kc(V ⊕W ).

Proof. Let t ∈ R and consider

(γt)v,w ∶= γv,w + te = c(v) + ic(w) + te ∶ π∗E0 → π∗E1.

Compute

(c(v) + ic(w) + te)2 == c(v)2 + ic(v)c(w) + tc(v)e + ic(w)c(v) − c(w)2 + itc(w)e + tec(v) + itec(w) + t2e2 =

= c(v)2 − c(w)2 − t2

and for t ≠ 0, this is strictly negative and hence (γt)v,w is invertible for all v,wand for all t ≠ 0. Therefore, γt is a homotopy to an acyclic K-cycle.

Proposition 10.5.9. Let M be a closed Riemannian manifold, E →M be a gradedCl(TM)-module and D be a Dirac operator. If E is inessential, then both, thetopological and the analytical index of D are trivial.

Proof. Lemma 10.5.8 and Example 10.5.3 prove that the symbol class σ(D) is zero,hence the topological index vanishes. Let D be a Dirac operator and let

B ∶= 1

2(D + eDe)

Since e anticommutes with Clifford multiplication and has e2 = −1, it followseasily that B has the same symbol as D. Moreover, Bι = −ιB is easily verified.As D is self-adjoint and e skew-adjoint, B is also self-adjoint. Hence B is a Diracoperator, and ind(D) = ind(B) by Proposition 10.3.3. For t ∈ R, B + ite is also aDirac operator, and ind(B) = ind(B + ite). Moreover Be = −Be and thus

(B + ite)2 = B2 + t2

and this is positive if t ≠ 0. Therefore, the kernel of B + ite is zero, and thusind(B + ite) = 0.

Proposition 10.5.10. Let V →X be an odd-dimensional oriented Riemann vectorbundle. Then each Cl(V )-module is inessential.

Proof. Let (v1, . . . , v2n+1) be an oriented orthonormal frame and put

Ω = inv1⋯v2n+1.

Using Lemma 4.1.2 and the Clifford identities, one proves that Ω does not dependon the choice of the oriented frame. It is easy to compute


Ω2 = −1; Ωc(v) = −c(v)Ω; Ωι = −ιΩ.Therefore, Ω exhibits E as an inessential Cl(V )-module.

Corollary 10.5.11. The Atiyah-Singer index theorem holds for oriented, odd-dimensional manifolds (and is of little interest).

We will now formulate the Atiyah-Singer index theorem for Dirac operators. LetV → X be an even dimensional oriented vector bundle on a compact space X andE be a Cl(V )-module. We define the characteristic class of E by

λ(E) ∶= (−1)nth−1(ch(abs(E))) ∈H∗(X).The multiplicativity of the Atiyah-Bott-Shapiro construction, of the Chern char-

acter and of the Thom isomorphism together yields the suggestive formula

λ(E ♯F ) = λ(E) × λ(F ).Using 10.5.3 and the cohomological reformulation of the K-theoretic index the-

orem, we arrive at the following formulation.

Proposition 10.5.12. Let M2n be a closed oriented manifold, E a Cl(TM)-moduleand D a Dirac operator. Then

top − ind(D) = ∫Mλ(E)Td(TM ⊗C).

Theorem 10.5.13. (Atiyah-Singer index theorem for Dirac operators) Let M2n bea closed oriented manifold and E a Cl(TM)-module.

10.6. The bundle modification theorem. We will now undertake the first bigstep towards the proof of the index theorem. Namely, we reduce the index theoremfor Dirac operators to the case of hypersurfaces, i.e. closed submanifolds M2n ⊂R2n+1. Here is the technical result.

Theorem 10.6.1. Let M be a closed oriented manifold, (E, ι, c) be a gradedCl(TM)-bundle and V →M be an oriented Riemann vector bundle of odd rank, andlet S →X be the sphere bundle of V . Then there exists a Cl(TS)-module (E′, ι′, c′)such that

top − ind(E) = top − ind(E′); ind(E) = ind(E′).

Corollary 10.6.2. The index theorem for hypersurfaces implies the index theoremfor all manifolds.

Proof. Let N2n be a closed oriented manifold. According to Whitney’s embeddingtheorem, we can find an embedding N → R2n+2m+1. Let V → N be the normalbundle of this embedding; then the sphere bundle S is a hypersurface. Let E → Nbe a Cl(TN)-module. By Theorem 10.6.1, we find a Cl(TS)-module E′, such that

ind(E) = ind(E′) = top − ind(E′) = top − ind(E)(the second equality is the index theorem hypersurfaces).

A large part of the proof of the bundle modification theorem can be carried outin a more general setting. We make the following assumptions.

186 JOHANNES EBERT

(1) G is a compact Lie group,(2) N a closed Riemannian manifold, with an isometric G-action.(3) F → N is a Cl(TN)-module, which is G-equivariant.(4) B is a G-equivariant Dirac operator on F .(5) G acts trivially on the kernel of B.(6) P →M is a smooth G-principal bundle over another manifold M .(7) q ∶ Q ∶= P ×G N →M is the induced fibre bundle.(8) The vertical tangent bundle of Q is TvQ ∶= ker(Tq) = P ×G TN , it inherits

a metric from TN . There is a splitting TQ = q∗TM ⊕ TvQ, q∗TM has ametric from M , and we pick a metric on Q such that the above splitting isorthogonal.

You should think of G = SO(2m + 1) and N = S2n. To find the appropriateexample of F , recall that for each even-dimensional oriented Riemann manifoldN , the exterior algebra ΛT ∗N is a Cl(TN)-module. We have two gradings: theeven/odd-grading ι and the grading τ , defined by the Hodge star. Recall theoperator D = d + d∗ and that

ind(D, ι) = χ(N); ind(D,τ) = sign(N).Moreover, the two grading operators ι and τ commute, so that η ∶= ιτ is a

Clifford-linear involution and commutes with D. Let Di be the restriction of D tothe (−1)i-eigenbundle of η. We get that

ind(D0, τ) + ind(D1, τ) = sign(N)and

ind(D0, τ)− ind(D1, τ) = ind(D0, τ)+ ind(D1,−τ) = ind(D0, ι)+ ind(D1, ι) = χ(N).

(the first equation is that reversing the grading reverses the sign of the index,the second equation is that on the ±-eigenbundle of η, the identity τ = ±ι holds).Now set

(10.6.3) F ∶= (Eig(η,+1), τ).

Since all data involved in this definition are invariant under orientation-preservingisometries, this Clifford module is G-equivariant if G acts on N by orientation-preserving isometries. If N = S2n, then the index of F is 1 (the Euler number is2 and the signature is 0). Moreover, the Dirac operator D on F is G-equivariantand its kernel can, by Hodge theory, be identified with a subspace of the cohomol-ogy of N . In particular, if N = S2n, then ker(D) is one-dimensional and, sinceSO(2m+ 1) is connected and acts trivially on the cohomology of S2n, has a trivialSO(2m + 1)-action.

Returning to the general situation, we form a Cl(TvQ)-module

F ′ ∶= P ×G Fand a Cl(TQ)-module

E′ ∶= q∗E ♯F ′.


Let us compute the topological index of the Clifford module E′. The computationis (all classes in sight have even degree and so they commute)

top − ind(E′) = ∫Qq∗(λ(E)Td(TM ⊗C))λ(F ′)Td(TvQ⊗C) =

∫Mλ(E)Td(TM ⊗C))λ(F ′)q!(λ(F ′)Td(TvQ⊗C)).

Lemma 10.6.4. If P →M is the frame bundle of the vector bundle V and F is asin 10.6.3, then

q!(λ(F ′)Td(TvQ⊗C)) = 1 ∈H∗(M).

Proof. We have computed the class λ(F ′)Td(TvQ ⊗ C) in the previous chapter.The answer was that

λ(F ′)Td(TvQ⊗C) = 1

2(e(TvQ) +L(TvQ)).

It follows that q!(e(TvQ)) = 2 = χ(S2n). Recall that cL(TvQ) is a polynomial inthe Pontrjagin classes of TvQ, hence L(TvQ) = L(TvQ ⊕ R). On the other hand,TvQ⊕R ≅ q∗V (a parametrized version of the fact that the tangent bundle of S2n

is stably trivial). This finishes the proof.

So if we pick F according to 10.6.3, we obtain a Clifford module E′ that hasthe same topological index as the original module E: For the computation of theanalytical index, we need to construct an actual operator. The construction willbe done in two steps. First, we construct a ”vertical operator B′ on F ′, which actsonly in the fibre direction and whose restriction to a fibre is the original operatorB. More precisely, we require

(1) Bv is an operator on sections of the bundle F ′ = P ×G F → Q = P ×G N .(2) Bv is formally self-adjoint, graded and of order 1.(3) Bv is C∞(M)-linear (so in particular not elliptic).(4) The symbol of Bv is given by smb1(Bv)(ξ, η) = ic(η). Here, we use the

splitting of the tangent bundle of Q and c is the Clifford multiplication ofTvQ on F ′.

(5) For each p ∈ P , let jp ∶ N → Q, x↦ [p, x], be the associated parametrizationof the fibre. There is a natural isomorphism j∗pF

′ = F , and hence a pullbackmap Γ(Q;F ′)→ Γ(N ;F ). We require that

Γ(Q,F ′) //

Bv

Γ(N ;F )

B

Γ(Q,F ′) // Γ(N ;F )

commutes.

From these conditions, one obtains that the kernel of Bv is the space of smoothsections of the bundle P ×G ker(B)→ N .

Lemma 10.6.5. Such a vertical operator Bv exists.

188 JOHANNES EBERT

Proof. We remark that the problem is local in M . First, if P =M ×G is the trivialbundle, the operator Bv is just the product of the operator B with the identity(it is obvious what this means in local coordinates). It remains to show that thisoperator Bv is invariant under change of frame, i.e. if g ∶M → G is such a functionand φg ∶ M ×N → M ×N is (m,n) ↦ (m,g(m)n), then φ∗g Bv = Bv φ∗g , whichfollows from the conditions (linearity over C∞(M) and G-equivariance of B). Sothe local operators defined by trivializations all agree.

If E →M is another vector bundle, we can form the operator Dv ∶= Bv ⊗ 1q∗E .Again, the C∞-linearity makes this unambigious. Now let E → M be a Cliffordbundle and D be a Dirac operator. The next step is to construct a ”horizontaloperator” Dh on sections of F ′ ⊗ F . The properties of Dh are

(1) Dh is formally self-adjoint, graded, order 1 and it anticommutes with Dv.(2) Dh +Dv is a Dirac operator on q∗E ⊗ F ′ (this fixes the symbol of Dh).(3) The restriction of Dh to

ker(Dv) = Γ(M ;E ⊗ ker(B)) = Γ(M ;E)⊗ ker(B)

(this needs the assumption that G acts trivially on the kernel of B)agrees with D ⊗ 1ker(B).

Lemma 10.6.6. Such a horizontal operator exists, after an appropriate choice ofD

Proof. Again, the problem is local, in two senses. More specifically, let (λi) be asquare-partition of unity (∑i λ2

i = 1) on M . Let Di be a Dirac operator on M andDh,i be such horizontal operators over the support of λi q. Then the operators

D =∑i

λiDλi; Dh =∑i

(q∗λi)Dh,i(q∗λi)

also satisfy these conditions. Furthermore, the conditions on D and Dh are in-variant under changes of frame. So it is enough to construct the horizontal operatorwhen P =M ×G, and this is easy: take the product Dh =D ⊗ 1N .

Now we assume that M is closed and compute the kernel of the Dirac operator

D′ =Dv +Dh

on q∗E ⊗ F ′. If D′u = 0, then

0 = ∥Dvu +Dhu∥2 = ∥Dvu∥2 + ∥Dhu∥2

since they anticommute and so

ker(D′) = ker(Dh) ∩ ker(Dv) = ker(Dh∣kerDv) = ker(D)⊗ ker(B).

The choice of the special operator on the sphere with one-dimensional kernelfinishes the proof of the bundle modification theorem.


10.7. Representation theory of Clifford algebras - algebraic Bott period-icity. By the result of the previous section, it is enough to prove the index theoremfor hypersurfaces. There are two main parts, one of algebraic nature and the otherone of analytic nature. The algebraic heart of the index theorem for Dirac opera-tors is the representation theory of the Clifford algebra. Here is the question thatwe will try to answer. Let M be a closed Riemann manifold and E → M be aCl(TM)-module with Dirac operator. Let W → X be a graded hermitian vectorbundle (i.e. a Clifford module over the zero bundle). Then the product E ⊗W isanother Cl(TM)-bundle. We denote any Dirac operator on E ⊗W by the sym-bol DW , the operator D twisted by W . This construction gives rise to two mapsK0(M)→ Z, namely

K0(M) ∋W ↦ abs(E ⊗W ) ∈Kc(TM) top−ind→ Zand

W ↦ ind(DW ) ∈ Z.By the index theorem, these maps of course agree. Now suppose, tentatively,

that E has the property that any other Clifford bundle on M is of the form E⊗W ,for some graded Hermitian bundle W . Then the index theorem on M asserts thattwo maps K0(M) → Z agree. If we know generators for K0(M), this can bechecked on finitely many examples. The question is whether any manifold admitssuch a Clifford module. This is not the case, there is an obstruction in H3(M ;Z)against this. Manifolds that do have this property are called Spinc-manifolds, andit turns out that hypersufaces are Spinc, for example the sphere. We will use ourknowledge of K0(S2n) (Bott periodicity!) and the Gauss-Bonnet theoem to provethe index theorem for the sphere. The full index theorem is proven by two moresteps: bordism invariance and bundle modification.

Let us begin with the work. The first step will be to determine all graded Clp,q-modules. We do it only in the complex case, which is easier than the real case. Thereal case is structurally similar, but more complicated.

Definition 10.7.1. Let V be a pseudo-euclidean vector space and (E, ι, c) be aCl(V )-module. We say that E is irreducible if E ≠ 0 and if E does not contain anyproper graded Cl(V )-submodule.

Definition 10.7.2. Let V → X be a pseudo-Riemannian vector bundle. ByMod(Cl(V )), we denote the category of all graded Cl(V )-modules. It is a linearcategory, enriched over Z/2-graded vector spaces.

Lemma 10.7.3. (Schur’s lemma) Let V be a pseudo-euclidean vector space and let

E0,E1 be irreducible complex Cl(V )-modules. Then Hom0(E0,E1) and Hom1(E0,E1)are at most one dimensional.

Proof. This is proven exactly as Schurs lemma in the representation theory ofgroups. Let f ∶ E0 → E1 be Cl(V )-linear and graded. Then ker(f) and Im(f)are submodules, and by irreducibility, it follows that f is zero or an isomorphism.So if there is any nonzero graded homomorphism, then E0 and E1 are isomorphic,and we can assume that E0 = E1 = E. Let λ ∈ C an eigenvalue. Then ker(f − λ) isa nonzero submodule, and hence ker(f − λ) = E, whence f = λ.

190 JOHANNES EBERT

An odd homomorphism f ∶ (E0, ι0) → (E1, ι1) can be interpreted as an evenhomomorphism (E0, ι0)→ (E1,−ι1). By the first part, the space of odd homomor-phisms is also at most one-dimensional.

It is possible that both spaces are one-dimensional, as we will show in an examplebelow.

Proposition 10.7.4. Any Clp,q-module is completely reducible, in other words itdecomposes as a direct sum of irreducibles.

Proof. Let E be a Clp,q-module and F ⊂ E be a submodule. We claim that F isalso a Clp,q-submodule. Cleary, F is invariant under the grading. So let v ∈ V ,x ∈ F and y ∈ F . Then

⟨c(v)x, y⟩ = ⟨x, c(v)∗y⟩ = ⟨x,−c(τv)y⟩since Clp,q → End(E) is a ∗-homomorphism. So F is also a submodule, and

this decomposes E as a direct sum of Clifford modules. Proceed by induction onthe dimension of E.

Theorem 10.7.5. (Algebraic Bott periodicity) Let V → X be a pseudo-Riemann

vector bundle. Then the tensor product with the Cl1,1-module S1,1 induces an equiv-alence of categories:

Mod(Cl(V ))→Mod(Cl(V ⊕R1,1)); E ↦ S1,1 ⊗E,which preserves direct sums and hence irreducibles and moreover inessential mod-ules.

Proof. This theorem holds over the reals. Recall that S1,1 is given by R2, with

ι = (1−1

) ; e = ( −11

) ; ε = ( 11

) .

Let (E,η, c) be a Cl(V )-module. Then S1,1 ⊗E is given by E ⊕E, with grading

(η −η) .

The Clifford action of v ∈ V is given by

(c(v) −c(v))

and the last two vectors act by the formulae for e and ε as above. One obtains Eback by restricting to the +1-eigenspace of the involution εe. It is an enlighteningroutine check that this correspondence gives indeed an equivalence of graded linearcategories.

Theorem 10.7.6. (Algebraic Bott periodicity, part II) Let V,W be two Riemannvector bundles. Then there is an equivalence of graded linear categories (here theuse of complex numbers is important):

Mod(Cl((V,+)⊕ (W,−)))→Mod(Cl(V ⊕W,+1)),which sends (E, ι, c) to (E, ι, c) (same grading), and the new Clifford action is

given by


c(v) = c(v); c(w) = ic(w).

Proof. This is trivial to check and only stated as a theorem because of its impor-tance.

Corollary 10.7.7. Mod(Cl(V )) ≅ Mod(Cl(V ⊕R2)).

This is the algebraic Bott periodicity in the complex case. In the real case, oneproves that

Mod(Cl(V ⊕R4,0)) ≅ Mod(Cl(V ⊕R0,4))using similar arguments. This proves the 8-periodicity. We can completely

describe now the complex modules over the Clifford algebra.

Example 10.7.8. A graded Cl0,0-module is nothing else than a graded complexvector space V = V0⊕V1. There are precisely two nonisomorphic graded modules inthis case. One is S0 ∶= (C,+1) (+1 is the grading), and the other one is (C,+1)op =(C;−1). It is clear that these two are not isomorphic. Furthermore, Hom0(S0,S0) =C and Hom0(S0,S0) = 0.

Example 10.7.9. There is one Cl1,0-module that is easy to describe. Let e ∈ Cl1,0

be the generator with e2 = −1. Let E be a graded module. Multiplication bye induces an isomorphism e ∶ E0 → E1. We identify these two part using thisisomorphism. Thus we can describe E as E0 ⊕E0 with grading and Clifford actiongiven by

ι = (1−1

) ; e = ( −11

) .

This module is irreducible iff dim(E0) = 1. Therefore, there is precisely one

irreducible Cl1,0-module. Both Hom0 and Hom1 are one-dimensional.

Theorem 10.7.10.

(1) Let n = 2m. Then there are precisely two isomorphism types of graded

irreducible complex Cl2m,0-modules S+2m and S−2m. Their dimension as acomplex vector space is 2m. They differ only by the sign of the gradingoperator. The total endomorphism space End(S±2m) is one-dimensional andeven. We can write S±2m+2 = S+2 ⊗ S±2m.

(2) Let n = 2m + 1. Then there is precisely one isomorphism class of graded

irreducible complex Cl2m+1,0-modules S2m+1. Its dimension is 2m, andEnd(S2m+1,0) is two-dimensional.

This follows from the two algebraic Bott periodicity theorems and the two ex-amples. One uses that the algebraic Bott periodicity preserves all structures insight.

Theorem 10.7.11. Let An be the monoid of equivalence classes of complex Cln,0-modules, modulo the inessential ones. Then An is a group and the Atiyah-Bott-Shapiro construction yields a ring isomorphism

A∗ ≅K−∗(∗).

The proof is an exercise.

192 JOHANNES EBERT

Theorem 10.7.12. Let E be a graded Cl2m,0-module and let S2m = S±2m. Then thenatural map

S2m ⊗HomCl2m,0(S2m;E)→ E; s⊗ f ↦ f(s)is a natural isomorphism of graded Cl2m,0-modules.

Proof. ”Natural”, ”Cl2m,0-linear” and ”graded” is trivial to check. To show that itis an isomorphism, we may decompose E as a sum of irreducibles. If E = S2m, thenHomCl2m,0(S2m;E) is an even line, and if E = S2m, it is an odd line. In both cases,the theorem is trivial to verify.

10.8. Spinc-manifolds and the index theorem on the sphere.

Definition 10.8.1. Let V → X be a Riemann vector bundle of rank n. A Spinc-structure on V is by definition a graded complex Cl(V ⊕R0,n)-module S which isfibrewise irreducible.

Remarks 10.8.2. You might have heard of a different definition of Spinc-structures,in terms of principal bundles. Of course both definitions agree. One can show thata vector bundle V →X admits a Spinc-structure if and only if it is orientable and ifa certain obstruction βw2(V ) ∈ H3(X,Z) is zero. Here w2(V ) ∈ H2(X;Z/2) is thesecond Stiefel-Whitney class of the bundle V and β ∶H2(X,Z/2)→H3(X;Z) is theBockstein operator (the connecting homomorphism in the long exact cohomologysequence associated with the coefficient sequence 0 → Z → Z → Z/2 → 0. One canalso show that all complex vector bundles (viewed as real vector bundles) are Spinc.

Remark 10.8.3. For even-dimensional vector bundles, we can use the algebraicBott periodicity and transform the Cl(V ⊕ R0,2n)-module S into a Cl(V )-moduleS. Similarly, for odd-dimensional bundles, we create an Cl(V ⊕R0,1)-module. It isoften convenient to work with these reduced modules.

First, we note the following corollary of Theorem 10.7.12.

Corollary 10.8.4. Let V →X be an even-dimensional Riemann vector bundle andlet S → X be a Spinc-structure on V (viewed as a Cl(V )-module). Let E → X beany Cl(V )-module bundle. Then E ≅ S ⊗HomCl(V )(S,E).

Proposition 10.8.5. A Spinc-structure on a vector bundle defines an orientation.

Proof. Let (v1, . . . , vn) be a local orthogonal frame of V . Consider the element

ω ∶= v1ε1⋯vnεn ∈ Cl(V ⊕R0,n).It is easy to see that ω commutes with ι and that ω anticommutes with Clifford

multiplication. Moreover

ω2 = 1

is quickly checked. Now the chirality operator is Γ = ιω. This is Cl(V ⊕ R0,n)-linear and even. By Schur’s lemma, Γ must be a scalar, and since Γ2 = 1, it mustbe ±1. We call (v1, . . . , vn) positively oriented if Γ = 1. We obtain a locall constantfunction Fr(V )→ ±1. As changing one of the vectors vi to its negative changes thesign of Γ, this function is not constant on the fibres of Fr(V ) and hence it reallygives an orientation.

Finally, we remark that passing to the opposite Spinc structure reverts the ori-entation of V .


Example 10.8.6. A trivial vector bundle X × Rn has a Spinc, namely X × Sn,n.Spinc-structures on V are in bijectio with Spinc-structures on V ⊕R.

Corollary 10.8.7. Let Mn ⊂ Rn+1 be an oriented hypersurface. Then M has aSpinc-structure.

Proof. We single out a specific construction. There exists a normal vector fieldν along M ; ν(x) is orthogonal to TxM and has length 1. This is because M isoriented. We obtain as isomorphism

TM ⊕R ≅M ×Rn+1; (v, t)↦ v + tν.Applying the algebraic Bott periodicity map to the Cl(TRn+1∣M⊕R0,n+1-module

M × Sn+1,n+1 yields the Spinc-structure on M .

Using this simple fact, we can prove the first instance of the index theorem.

Theorem 10.8.8. The Atiyah-Singer index theorem for Dirac operators is true forthe sphere.

Proof. We use an argument that is similar to the proof of the Riemann-Roch the-orem, given last term.

The result is true for S0, for trivial reasons. Let us first say how we use theSpinc-condition. Let M be a Spinc manifold, with spinor bundle S. We define amap

J ∶K0(M)→ Z;V ↦ ind(DS⊗V ).This is well-defined, since the symbol of DS⊗V is uniquely determined, and since

the index only depends on the symbol. By Corollary 10.8.4, each graded Diracoperator is of the form (on the symbol level) DS⊗V . If we know generators forK0(M)⊗Q for which the index formula is true, it follows for all V and hence for allDirac operators. By Bott periodicity, K0(S2n) ≅ Z2. We know two operators on thesphere for which the index theorem is true. The first is, by the Gauß-Bonnet-Cherntheorem, the Euler characteristic operator. We calculated the index last term, usingHodge theory, and computed the topological index. Both answers agree.

The second operator is the signature operator! The signature of the sphere (theanalytical index) is evidently zero, since Hn(S2n) = 0 (we also used Hodge theoryhere!). On the other hand, we showed that the topological index of the signatureoperator is the L-genus, an integral over the Pontrjagin classes. But the sphere isstably parallelizable and therefore has trivial Pontrjagin classes, and the topologicalindex is also zero.

To make the argument in any sense conclusive, we need to know the following.Both operators are of the form DS⊗E , for some elements E ∈K0(S2n), and we needto know that the elements Eeul and Esign are linearly independent. Fortunately,we do not need to determine these twisting bundles. The point is that we have asecond homomorphism

I ∶K0(S2n)→K0c (TS2n)→K0

c (R2n) ≅ Z.The first map sends a vector bundle V to the symbol class σ(DV ) of the twisted

Dirac operator, the second map is given by restriction to the tangent space at thebase point (which one does not matter, as the sphere is connected). The third mapis the inverse to the Bott map. We calculated that

194 JOHANNES EBERT

J(Eeul) = 2; J(Esign) = 0; I(Esign) = 22n; I(Eeul) = 0.

Therefore, the two elements Eeul,Esign ∈K0(S2n)⊗Q are linearly independent.This completes the proof.

10.9. The bordism theorem. It remains to show the index theorem for arbitraryhypersurfaces M2n ⊂ R2n+1, generalizing the case M = S2n. Let us first explain theidea. Without loss of generality, we can restrict to connected hypersurfaces (bothindices are additive with respect to disjoint union) which are contained in the openunit ball. We need a basic fact from algebraic topology.

Lemma 10.9.1. Let M2n ⊂ R2n+1 be a connected closed hypersurface. Then Mdivides Rn into two connected components.

Proof. We use the long exact sequence

H∗c (R2n+1 −M) // H∗

c (R2n+1)

xxH∗(M)

[+1]

gg

Using that H2n(M) ≅ R and the known computation of H∗c (R2n+1), we obtain

that dimH2n+1c (R2n+1−M) = 2. By Poincare duality, this implies that dimH0(R2n+1−

M) = 2.

Let M ⊂ R2n+1 be contained in the open unit ball. According to Lemma 10.9.1,we can write the closed unit disc as a union

D2n+1 =W0 ∪M W1.

The manifold W0 is the inner part, and W1 the outer part. Both are bordisms.

Definition 10.9.2. A bordism W ∶ M0 M1 between two closed manifolds is acompact manifold W ; together with a decomposition

∂W = ∂0W∐∂1W

and diffeomorphisms φi ∶ ∂iW ≅Mi.

There is the following extra structure. From differential topology, we import thefollowing technical result.

Theorem 10.9.3. Let W ∶ M0 M1 be a bordism. Then there exist collars, i.e.open neighborhoods Ui ⊃ ∂iW and diffeomorphisms

U0 ≅M0 × [0, ε); U1 ≅M1 × (1 − ε,1]extending the given diffeomorphisms φi.

We let t ∶W → [0,1] be a smooth function that extends the projections onto thelast coordinates on Ui. The normal vector field is

ν = ∂

∂t.

It is defined on all of W , but we use it only on the collar. The vector field pointsinto the bordism along ∂0W and out of the bordism at the other boundary.


We consider Riemann metrics on bordisms, but they need to have a specialstructure, namely, if gi are Riemann metrics on Mi, then the metric h on W needsto have the form

h = gi + dt2

on the collar. The normal vector field induces vector bundle isometries

ηi ∶ TMi ⊕R ≅ TW ∣Mi ; (v, x)↦ v + xν.Let E → W be a Cl(TW ⊕ R0,1)-module. We require that E is a product on

the collars. By the algebraic Bott periodicity, we obtain Cl(TMi)-modules ∂iE, byapplying the following functors

Mod(Cl(TW⊕R0,1)) res→ Mod(Cl(TW⊕R0,1∣Mi))ηi≅ Mod(Cl(TMi⊕R1,1))

ABP≅ Mod(Cl(TMi)).

We call the image of E the boundary reduction of E and denote it by ∂iE. Inparticular, a given Spinc-structure SW on W induces Spinc-structures SMi on Mi.If V →W is a vector bundle, then the boundary reduction of SW ⊗V is SMi⊗V ∣Mi .

In the definition of a bordism, there is an inherent sense of direction. But letme point out that the direction by itself has no geometric significance. We caneasily view a bordism W ∶ M0 M1 as a bordism W ∶ ∅ M0∐M1. The effectis that we change the sign of ν along the boundary piece M0. It is not hard tocheck that this implies that the boundary reduction of a Clifford module E →W isEop0 ∐E1. According to the orientation convention for boundaries, this also reversesthe orientation of M0, but that aspect does not play a big role for us. We are readyto state the bordism theorem.

Theorem 10.9.4. Let W ∶M0 M1 be a bordism and let E →W be a Cl(TW ⊕R0,1)-module. Let ∂iE be the boundary reductions of E to Mi. Then

(1) The topological index is bordism invariant: top − ind(∂0E) = top − ind(∂1E)and

(2) the analytical index is bordisminvariant as well: ind(∂0E) = ind(∂1E).

The bordism invariance of the analytical index is the hard part and we followan ingenious argument given by Higson. Before we attack this problem, we givethe relatively easy proof of the invariance of the topological index and then explainwhy the bordism theorem implies the index theorem.

Proof of Theorem 10.9.4, part 1. Bending the bordismW to a bordism ∅M0∐M1

changes the boundary reduction ∂0E to (∂0E)op. Since passing to the opposite Clif-ford module changes the signs of both indices, it is enough to prove the theoremfor a bordism from ∅ to M =M0∐M1. Let E →W be a Cl(TW ⊕R0,1)-module.The claim is that

top − ind(∂E) = 0.

The cohomological formula for the topological index is

top − ind(∂E) = ∫Mλ(∂E)Td(TM ⊗C).

196 JOHANNES EBERT

Let j ∶ M → W be the inclusion. We will show that both factors under theintegral lie in the image of j∗ ∶H∗(W )→H∗(M). By Stokes’ theorem, the integralis therefore 0. The Todd class is

Td(TM ⊗C) = Td((TM ⊕R)⊗C) = Td(TW ∣M ⊗C) = j∗Td(TW ⊗C).

By algebraic Bott periodicity, we can write ∂E ⊗ S1,1 = E∣M . Since λ(S1,1) = ±1(this is the computation of the Chern character of the tautological bundle on CP1

again, in disguise!), we find by the multiplicativity of λ:

±λ(∂E) = λ(∂E ⊗ S1,1) = λ(j∗E) = j∗λ(E).This completes the proof.

Proof of the Atiyah-Singer index theorem, assuming the bordism theorem. LetM ⊂R2n+1 be a hypersurface contained in the open unit disc and dividing the disc intotwo bordisms W0 ∶ ∅M , W1 ∶M S2n. Let S be a spin structure on R2n+1 andlet SM be the boundary reduction to M . Consider the maps

I, J ∶K0(M)→ Z; V ↦ ind(SM ⊗ V ); top − ind(SM ⊗ V ).The index theorem says that both maps are equal. Let V ′ → W1 be a vector

bundle. Then

ind(SM⊗V ′∣M) = ind(SS2n⊗V ′∣S2n) = top − ind(SS2n⊗V ′∣S2n) = top − ind(SM⊗V ′∣M).

The first equation is the analytical part of the bordism theorem, the secondequality is the index theorem on the sphere and the third equation the topologicalpart of the bordism theorem. In a similar, but easier, way, one sees that if V ′ →W0

is a vector bundle, then

ind(SM ⊗ V ′∣M) = top − ind(SM ⊗ V ′∣M) = 0

(technically, this is the index theorem on ∅, which is completely trivial). So wefind that I = J on the image of the sum of the restriction maps

K0(W0)⊕K0(W1)→K0(M).The proof will be completed if we show that the above map is surjective. But

this map fits into the Mayer-Vietoris sequence in K-theory (D is the closed unitdisc):

K0(D) // K0(W0)⊕K0(W1) // K0(M)

K−1(M)

OO

K−1(W0)⊕K−1(W1)oo K−1(D)oo

Since K−1(D) = 0, the sum of the restriction maps is surjective, and the proofis complete. We remark that for this step the full glory of the Bott periodicitytheorem is needed again: the long exact sequence can easily be extended to theleft, but extending it to the right was nontrivial.


10.10. The reformulation of the bordism theorem. To state the precise an-alytical version of the bordism theorem, we need a slight reformulation of the al-gebraic Bott periodicity. Let V → X be a Riemannian vector bundle. Let E be aCl(V ⊕R1,1)-module, with Clifford actions e and ε by the last basis vectors. Theimage of E under algebraic Bott periodicity is

Eig(εe,+1),with the grading and the Cl(V )-action restricted. Consider the endomorphism

iει of E instead. It is easy to see that iει is Cl(V )-linear and an involution. Hencethe eigenspace

Eig(iει,+1)is an (ungraded) Cl(V )-module. A grading on Eig(iει,+1) is given by ie.

Lemma 10.10.1. There is an isomorphism of graded Cl(V )-modules (Eig(εe,+1), ι)and (Eig(iει,+1), ie).

Proof. This is straightforward if one writes E = F ⊗ S1,1, according to algebraicBott periodicity.

What did we gain by this reformulation? Consider a bordism W ∶M0 M1 anda Cl(TW ⊕R0,1)-module E →W . By the above lemma, we can write the boundaryreduction ∂iE as Eig(iει,+1), with grading given by ie (this is the Clifford multi-plication by the normal vector field ν). The point now is that the endomorphism

iει is defined on all of W ! Let E ∶= Eig(iει,+1)→W , this is an ungraded Cl(TW )-module (this is the only place where we consider ungraded Clifford modules). Over

∂W , E has the grading σ = ic(ν), given by Clifford multiplication with the normalvector field (there is no need to discuss the two components of ∂W separately).

Now assume that Bi is a graded Dirac operator on ∂iE, i.e. on E, graded withrespect to σ. Now consider the operator

D ∶= iσ ∂∂t

+Bion the collars. It is quickly checked that this is an ungraded Dirac operator

on E∣U . By the general existence lemma for Dirac operators, we can extend theoperator D over all of W .

The general logic of the proof of the bordism theorem is that D is ”responsible”for the indices of B0 and B1 being equal. Conceptually, D should yield a homo-topy between the Fredholm operators B0 and B1. As it stands, this statement isnonsensical, not last because the two operators live in different Hilbert spaces.

It might seem that we need to discuss boundary value problems for elliptic op-erators on the manifold W with boundaries. We will not do this. Instead, we doanalysis on noncompact manifolds. Consider the elongation W of W , which is thenoncompact manifold

W ∶=M0 × (−∞,0] ∪M0 W ∪M1 M1 × [1,∞).We extend the Riemann metric on W to all of W , using that the metric was

cylindrical on the collars. Moreover, we extend the ungraded Cl(TW )-module to

all of W , as a product. Cleaning up notation, we finally formulate the preciseassumptions.

198 JOHANNES EBERT

Assumptions 10.10.2. Let W d be a Riemannian manifold, equipped with aproper smooth map t ∶ W → R. There are a0 < a1 ∈ R and closed manifoldsMi, such that t−1(−∞, a0] = M0 × (−∞, a0] (of course, we mean ”isomorphic” in-stead of ”equal”). The Riemann metric on t−1(−∞, a0] is of the form g0 + dt2, fora Riemann metric g0 on M0. Let E →W be an ungraded Cl(TW )-module, whichis a product on t−1(−∞, a0]. Assume that there is an grading involution σ0 on theCl(TM0)-module E0 = E∣M0 . Let B0 be a Dirac operator on E0, which is graded,i.e.

B0σ0 + σ0B0 = 0.

Let D be an ungraded Dirac operator on E →W , such that

D = iσ0∂

∂t+B0

on t−1(−∞, a0]. An analogous structure is assumed to hold on the other end, i.e.t−1[a1,∞).

The version of the bordism theorem 10.9.4 that we actually prove is

Theorem 10.10.3. Under the assumptions 10.10.2, the analytical indices of B0

on M0 and B1 on M1 are equal.

10.11. Elliptic analysis on manifolds with cylindrical ends. We work underthe assumptions 10.10.2; and the first big task is to extend the elliptic regularitytheory to the more general situation of noncompact manifolds. Let us define theSobolev space on the noncompact manifold W . Before we do this, let us introducesome notation. Let u ∈ Γc(W ;E) be a compactly supported section. Due to thecylindrical structure of W at infinity, we can view the restriction of u to the endsas functions

u ∶ (−∞, a0]→ Γ(M0;E0).Given any function µ on R, we can look at the composition µ t ∶ W → R. We

will keep the notation simple by denoting this pulled back function also by µ. TheHilbert space of sections of E → W is defined as usual: it is the completion ofΓc(W ;E) with respect to the norm

∥u∥2H ∶= ∫

W∣u(x)∣2dx.

We use the symbol H to distinguish it notationally from the Hilbert spacesL2(Mi,Ei) of sections on lower-dimensional manifolds.

Definition 10.11.1. The Sobolev space W of order 1 on W is the completionof Γc(W ;E) with respect to the norm defined by the following procedure. Letµ+, µ−, µ0 be smooth functions R→ [0,1], such that µ+ + µ− + µ0 = 1 and such that

supp(µ−) ⊂ (−∞, a0); supp(µ+) ⊂ (a1,∞); µ0 ∈ C∞c (R)

(note that this implies that µ+ is equal to 1 for t >> 0, µ− is equal to 1 for t << 0and µ0∣[a0,a1] = 1). Let U ⊂W be relatively compact and containing the support ofµ0 (= µ0 t). For u ∈ Γc(W ;E), we define


∥u∥2W ∶= ∫

a1

−∞∥µ−(t)u(t)∥2

W 1(M0,E0)dt+∫+∞

a0∥µ+(t)u(t)∥2

W 1(M1,E1)dt+∥µ0u∥2W 1(U).

We will not use Sobolev spaces of order different from 1.

Lemma 10.11.2. The equivalence class of the norm on W does not depend on thechoice of the functions µ+,−,0. The operator D on Γc(W ;E) extends to a boundedoperator D ∶W →H.

The proof is an easy adaption of the techniques we have seen in the last term andleft to the reader. The canonical expectation you might have is that we proceedto prove that the operator D is Fredholm. However, it turns out that this is falsewithout a further assumption.

Assumption 10.11.3. The operator D is called invertible at infinity if the oper-ators B0 and B1 on Mi are invertible.

You should not be bothered by the fact that we are ultimately interested in asituation where the operators Bi are not invertible. The main result of this sectionis

Theorem 10.11.4. If the operator D is invertible at infinity, then D ∶W → H isFredholm, and Im(D) = ker(D).

The assumption on invertibility at infinity is necessary for the theorem to hold.Here is a minimal counterexample.

Example 10.11.5. Let W = R, E = C, σ = 1, B = 0 and let D = i ∂∂t

. We claimthat D ∶ W → H is not Fredholm. The Sobolev norm of a function u ∈ C∞

c 8R) isgiven by

∥u∥2W ∶= ∫

+∞

−∞∣u(t)∣2 + ∣u′(t)∣2dt = ∥u∥2

L2 + ∥u′∥2L2 .

Assume that D is Fredholm. Since D is clearly injective, this means that D isa linear homeomorphism D ∶ W → Im(D) ⊂ L2 (by the closedness of the range ofFredholm operators and the open mapping theorem). In particular, there existsc > 0 such that for all u ∈W, we have

(10.11.6) c∥u∥2W ≤ ∥Du∥2

L2 .

We can choose c < 1. For s > 0, let us(t) ∶= u(st). A simple computation showsthat

∥us∥2L2 =

1

s∥u∥2

L2 ; ∥(us)′∥2L2 = s∥u′∥2

L2 .

When inserted into the estimate 10.11.6, we obtain

c(1

s∥u∥2

L2 + s∥u′∥2L2) = c(∥us∥2

L2 + ∥(us)′∥2L2) ≤ ∥(us)′∥L2 = s∥u′∥L2

or

1

s∥u∥2

L2 ≤1 − ccs∥u′∥L2

for all s > 0. For s→ 0, this is a contradiction.

200 JOHANNES EBERT

When s becomes small, us becomes more and more ”spread out”. This showsthat we have a problem at infinity, and intuitively, invertibility at infinity helps tocontrol the functions far outside.

The workhorse will be a version of Garding’s inequality.

Proposition 10.11.7. If D is invertible at infinity, there exists C ≥ 0 such thatfor all u ∈W, we have

∥u∥W ≤ C(∥µ0u∥L2(U) + ∥Du∥L2).

Proof. Recall that the proof of the usual Gardings inequality was done by provingit locally and then patching the pieces together. The same pattern applies here;the details how to patch are left to the reader. If u is supported inside U , theusual Garding’s inequality applies. The new ingredient is to study what happenson the cylindrical ends. So we may assume that W =M ×R and that D = iσ∂t +B;Bσ+σB = 0 and that B is invertible. Since B is invertible, there is a constant c > 0such that ∥Bv∥L2(M) ≥ c∥v∥W 1(M) for all v ∈W 1(M). Let u ∈W and estimate

∥u∥2W = ∫

+∞

−∞∥u(t)∥2

1+∥u′(t)∥20dt ≤ ∫

+∞

−∞

1

c2∥Bu(t)∥2

0+∥u′(t)∥20dt ≤ C ∫

+∞

−∞∥Bu(t)∥2

0+∥u′(t)∥20dt.

On the other hand, denoting by ⟨, ⟩ the L2-inner product on M ×R, we obtain

∥Du∥2L2 = ⟨Bu,Bu⟩ + ⟨u′, u′⟩ + ⟨i∂tσu,Bu⟩ + ⟨Bu, i∂tσu⟩.

The last two summands cancel out:

⟨i∂tσu,Bu⟩ + ⟨Bu, i∂tσu⟩ = ⟨(Bi∂tσ + σi∂tBu,u⟩ = 0,

using that i∂t is formally self-adjoint, that B and σ commute with ∂t and thatBσ + σB = 0. Altogether, we get

∥u∥2W ≤ C ∫

+∞

−∞∥Bu(t)∥2

0 + ∥u′(t)∥20dt = C∥Du∥2

L2 .

Proof of Theorem 10.11.4. By the abstract closed range lemma and by Rellich’stheorem, we find that D has finite dimensional kernel and closed image. It remainsto show that Im(D) = ker(D). The inclusion ker(D) ⊂ Im(D) is clear, and weonly have to show the opposite (which in particular, since the kernel is finitedi-mensional, implies the Fredholm property). Let v ∈ Im(D); this condition can bestated as

⟨Du, v⟩ = 0

for all u ∈ Γc(W,E). Suppose we know that v ∈ W, so in particular Dv ∈ Hexists. We have to prove that ⟨u,Dv⟩ = 0 for all u ∈ Γc. For ε > 0, pick a smooths ∈ Γc with ∥v − s∥W ≤ ε. Then

∣⟨u,Dv⟩∣ ≤ ∣⟨u,Ds⟩∣+∣⟨u,D(v−s)⟩∣ ≤ ∣⟨Du, s⟩∣+∥u∥H∥D(v−s)∥H ≤ ∣⟨Du, s⟩∣+Cε∥u∥H.On the other hand

∣⟨Du, s⟩∣ = ∣⟨Du, s−v⟩∣+∣⟨Du, v⟩∣ = ∣⟨Du, s−v⟩∣ ≤ ∥Du∥H∥s−v∥H ≤ ∥Du∥H∥s−v∥W ≤ ε∥Du∥H.


So ⟨u,Dv⟩ = 0, and this was to be shown.It remains to prove that v ∈W, which will be accomplished in two steps. First,

we prove that v is locally in W, in other words, for each compactly supportedsmooth function g, gv ∈W. This amounts to exerting local control over v. The laststep is to control the behaviour of v at infinity.

So assume that g is supported in the relatively compact subset U ⊂W . Let Fsbe a family of Friedrich’s mollifiers. We claim that ∥Fs(gv)∥W is globally (in s)bounded. By Proposition 3.4.3, this will prove that gv ∈W. By Gardings inequality(the compactly supported version of which suffices for this argument), we estimate

∥Fsgv∥W ≤ C(∥Fsgv∥H + ∥DFsgv∥H).Since the operator norm of Fs ∶H →H is bounded, we get that

∥Fsgv∥H ≤ C∥gv∥H ≤ C ′∥v∥H.To estimate the second summand, we compute, for u ∈ Γc,

∣⟨u,DFsgv⟩∣ = ∣⟨FsDu, gv⟩∣ ≤ ∣⟨[Fs,D]u, gv⟩∣ + ∣⟨DFsu, gv⟩∣.By Friedrichs’ lemma, the operator norm of [D,Fs] ∶H →H is globally bounded,

and so

∣⟨[Fs,D]u, gv⟩∣ ≤ C∥u∥H∥gv∥H ≤ C ′∥u∥.Moreover

∣⟨DFsu, gv⟩∣ = ∣⟨gDFsu, v⟩∣ ≤ ∣⟨[g,D]Fsu, v⟩∣ + ∣⟨D(gFsu), v⟩∣ = ∣⟨[g,D]Fsu, v⟩∣because v is orthogonal to Im(D). But [g,D] is compactly supported and of

order 0 and so

∣⟨[g,D]Fsu, v⟩∣ ≤ ∥[g,D]∥H,H∥Fs∥H,H∥u∥H∥v∥H.So altogether, we have a bound

∣⟨u,DFsgv⟩∣ ≤ C∥u∥H∥v∥Hand therefore a global bound on ∥DFsgv∥H and hence on ∥Fsgv∥W . It follows

that gv ∈W. So far, the assumption on invertibility at infinity was not used.For the control at infinity, pick a sequence gn of compactly supported functions

on R, with the following properties:

(1) gn∣[−n,n] ≡ 1,

(2) ∣(gn)′(t)∣ ≤ 1n

which, when the functions gn are transplanted to W , implies that

(1) For all u ∈H: ∥gnu − u∥H → 0 as n→∞ and(2) The operator norm ∥[D,gn]∥H,H → 0 as n→∞.

Let v ∈ Im(D); and we just showed that gnv ∈ W. We claim that gnv is aW-Cauchy sequence. By Gardings inequality, we obtain

∥(gn − gm)v∥W ≤ C(∥(gn − gm)µ0v∥H + ∥D(gn − gm)v∥H).The first summand is zero for n,m sufficiently large. Let u ∈ Γc be arbitrary.

Then

202 JOHANNES EBERT

∣⟨u,D(gn − gm)v⟩∣ = ∣⟨(gn − gm)Du, v⟩∣ ≤ ∣⟨[gn − gm,D]u, v⟩∣ + ∣⟨D(gn − gm)u, v⟩∣.The first equation is legal since gnv ∈ W. Now ∣⟨D(gn − gm)u, v⟩∣ = 0 as v ∈

Im(D). Moreover,

∣⟨[gn − gm,D]u, v⟩∣ ≤ ∥[gn − gm,D]∥H,H∥v∥H∥u∥H.Altogether, this shows that ∥(gn − gm)v∥W converges to zero as n,m → ∞. Thusgnv is a Cauchy sequence, and the limit must be v, so v ∈W.

Corollary 10.11.8. The operator (D ± i) ∶ W → H is invertible, without the as-sumption on invertibility at infinity.

The reader who is familiar with the theory of unbounded operators will noticethat Corollary 10.11.8 states that the operator D ∶ W → H is selfadjoint in thesense of unbounded operator theory.

Proof. We consider the operator

D = ( D − iD + i )

which over the ends has the form

D′ = ( D − iD + i ) = i( σ

σ)∂t + ( B − i

B + i ) =∶ iσ′∂t +B′.

It is obvious that σ′ is an involution that anticommutes with B′ and that

(B′)2 = (B2 + 1

B2 + 1) ≥ 1

so that B′ is invertible. In other words, the assumptions of our previous analysisare fulfilled. We conclude that D′ is Fredholm with Im(D′) = ker(D′). if we canshow that the kernel of D′ is trivial, it follows that D′ is invertible and hence thatD ± i is invertible. Assume that D′u = 0, with u = (u0, u1). A straightforwardcalculation shows that

0 = ⟨D′u,D′u⟩ = ∥Du0∥2H + ∥Du1∥2

H + ∥u0∥2H∥u1∥2

Hand this means that u0 = u1 = 0.

We cannot expect to get away with the proof of the index theorem without afurther explicit computation of an index (so far, we needed the computation of theindex on one Toeplitz operator for the proof of the Bott periodicity theorem andthe computation of the index of the signature and Euler characteristic operator ona sphere). We do this index computation now.

Theorem 10.11.9. (The baby spectral flow theorem) Let M be a closed manifoldand E →M a graded Cl(TM)-bundle, with grading σ and let B be a graded Diracoperator, not necessarily invertible at infinity. Let ψ ∶ R → R be a smooth functionsuch that ψ ≡ −1 near −∞ and ψ ≡ 1 near +∞. Let D = iσ∂t +B on M ×R, this isan ungraded formally selfadjoint operator from Γ(M ×R;E ×R) to itself. Then the(ungraded) operator (D − iψ) ∶W →H is Fredholm and has index


ind(D − iψ) = − ind(B,σ).

Proof. We consider the operator

D′ ∶= ( D + iψD − iψ ) = i( σ

σ)∂t + ( B + iψ

B − iψ ) =∶ iσ′∂t +B′.

Again, σ′ and B′ anticommute, B′ is formally selfadjoint. Moreover, near ±∞,we have

B′ = ( B ± iB ∓ i )

and this is invertible. So in particular, D′ satisfies the assumptions of our anal-ysis. Hence D − iψ is Fredholm, and Im(D − iψ) = ker(D + iψ). In other words

ind(D − iψ) = dim ker(D − iψ) − dim ker(D + iψ).We claim that there is an isomorphism

ker(D − iψ)⊕ ker(D + iψ) ≅ ker(B)of graded vector spaces (the kernel ker(D + iψ) is in odd degree, and the kernel

ker(D − iψ) in even degree). This will finish the proof. Let ε ∈ ±1 and u ∈ ker(D −εiψ). Then (∂ ∶= ∂t)

0 = ⟨(D − εiψ)u, (D − εiψ)u⟩ =⟨Bu,Bu⟩ + ⟨(iσ∂ − iεψ)u, (iσ∂ − iεψ)u⟩ + ⟨Bu, (iσ∂ − iεψ)u⟩ + ⟨(iσ∂ − iεψ)u,Bu⟩ =

∥Bu∥2H + ∥(iσ∂ − iεψ)u∥2

H + ⟨Bu, (iσ∂ − iεψ)u⟩ + ⟨(iσ∂ − iεψ)u,Bu⟩.

(10.11.10)

But

⟨Bu, (iσ∂ − iεψ)u⟩ + ⟨(iσ∂ − iεψ)u,Bu⟩ = ⟨u; (Biσ∂ −Biεψ + iσ∂B + iεψB)u⟩by the various adjunction relations. Moreover

Biσ∂ −Biεψ + iσ∂B + iεψB = Biσ∂ + iσ∂B = i∂(Bσ + σB) = 0

since B and σ commute with ∂ and since B and σ anticommute. Therefore, thelast two summands in 10.11.10 add to zero. This means that the function u(t) liesin ker(D − εiψ) if and only if

Bu(t) = 0 and iσu′(t) − iεψu(t) = 0.

In other words, u is a function into the finite-dimensional vector space ker(B)and satisfies the ODE iσu′(t)−iεψu(t) = 0. We look at the finite-dimensional vector

space ker(B) and write σ = (1−1

). We have to study the ODE

i(x′(t)

−y′(t)) − iεψ (x(t)y(t)) .

It is important that this are two uncoupled equation, one for x and one for y.The two equations are

204 JOHANNES EBERT

x′(t) − εψ(t)x(t) = 0; y′(t) + εψ(t)y(t) = 0.

We have to find the space of L2-solutions to these two equations. Let us assumethat ε = +1. On the part where ψ is constant, the solutions are given by

x(t) =⎧⎪⎪⎨⎪⎪⎩

c1et t >> 0

c0e−t t << 0.

Such a function can only be L2 if c0 = c1 = 0. By uniqueness of solutions toODE, this shows that x = 0. On the other hand, the solution to the equation for yis given (near infinity) by

y(t) =⎧⎪⎪⎨⎪⎪⎩

c1e−t t >> 0

c0et t << 0.

This is in L2 for an arbitrary choice of ci. Therefore, the solution to the equationy′(t) + ψ(t)y(t) = 0, for any initial value y(0), is in L2. Summarizing, we obtainthat

ker(D − iψ) = ker(B)1 = Eig(σ∣ker(B);−1).For ker(D + iψ), the argument is the same, up to a sign change, and yields

ker(D + iψ) = ker(B)0 = Eig(σ∣ker(B);+1).

Let us explain the name of the theorem. We considered the curve At ∶= −iσB +σψ(t) of self-adjoint differential operators on M and computed the index of theoperator ∂t+At. For t near infinity, the operators At are invertible, and the theoremsays that the index of ∂t + At counts the number of eigenvalues of At that crosszero (with multiplicity). This number is called ”spectral flow” of the family At,and the theorem says that the index of D equals the spectral flow of the familyAt. This result holds for much more general families of self-adjoint differentialoperators. The proof above used the special simple form of the family and cannotbe generalized easily.

10.12. The proof of the index theorem. We now are very close to our goal.Recall that we consider a manifold W 2n+1 with one cylindrical end M × [a,∞). Wehave an ungraded vector bundle E →W , which is a product on the end. There isa grading σ of E∣M , and a Dirac operator B with Bσ + σB = 0 on M . Moreover,we have an ungraded Dirac operator D on W such that D = iσ∂t +B on the end.We want to prove that ind(B,σ) = 0. We have proven that the operators (D ± i)are invertible. Let us form the Cayley transform

U = (D − i)(D + i)−1.

This is a unitary operator on the Hilbert space H (and it is invertible).

Proposition 10.12.1.

(1) For each compactly supported function f ∈ C∞c (W ); the operators f(D±i)−1

and (D ± i)−1f are compact.


(2) If h is a function on W such that dh is compactly supported (i.e., h islocally constant outside a compact subset), then [h, (D ± i)−1] is compact.

Proof. Let U ⊂ be relatively compact and contain the support of f . Then f(D±i)−1

is the composition

H(D±i)−1→ W f→W(U)→H.

The third map is compact by Rellich’s theorem, and thus f(D± i)−1 is compact.For the other composition, use that (f(D ± i)−1)∗ = (D ∓ i)−1f and that passing tothe adjoint preserves compactness.

For the second part, we abbreviate Q = (D ± i) for readability. It is easy tocompute that

−[Q−1, h] = Q−1[Q,h]Q−1.

Now [Q,h] is an order 0 operator and has compact support. So, by the first partof the Proposition, [Q,h]Q−1 is compact, and hence [Q−1, h] is compact.

We inch towards the central construction of the proof. Write

R = −2i(D + i)−1 so that U = 1 +R.

Definition 10.12.2. Let µ be a function on W such that µ2 − µ has compactsupport (in other words, outside a compact subset of W , the function µ is either 0or 1). We define a new operator Pµ ∶H →H by

Pµ = 1 + µR = (1 − µ) + µU.

The operator Pµ is an interpolation between the identity (where µ = 0) and theoperator R (where µ = 1).

Proposition 10.12.3.

(1) The operator Pµ is Fredholm.(2) ind(Pµ) = − ind(P1−µ).(3) If the support of µ − λ is compact, then ind(Pµ) = ind(P1−µ).

Proof. Let us compute

PµP1−µ = (1+µR)(1+(1−µ)R) = 1+R+µR(1−µ)R = 1+R+µ[R, (1−µ)]R+µ(1−µ)R2 ∼ 1+R = U.Here, ∼ means equality modulo compact operators and it holds by Proposition

10.12.1. So, from Atkinson’s theorem, we conclude that Pµ is Fredholm and thatind(Pµ) = − ind(P1−µ).

Similarly, if µ and λ are two such function and if the support of µ−λ is compact(in other words, they agree outside a compact set), then

Pµ − Pλ = (µ − λ)R ∼ 0.

Therefore, the difference is compact and so the indices are equal.

We can now finally prove the bordism theorem. For simplicity, we assume thatW0 ∶ ∅M is a nullbordism, and that the operators D on W0 and B on M are asbefore. We form the elongation W = W0 ∪M W1, W1 ∶= M × R. The manifold W

206 JOHANNES EBERT

comes with a proper map t ∶W → R sich that W0 = t−1(−∞,0] and such that W1 isthe cylinder.

Let µ ∶ R → [0,1] be a function such that µ ≡ 0 on a neighborhood of (−∞,0]and such that µ ≡ 1 near +∞. As usual, pull back the function µ to W . Byconstruction, the function 1 − µ ∶ W → [0,1] has compact support. Therefore, byProposition 10.12.3

ind(Pµ) = ind(P1) = ind(1 +R) = ind(U) = 0.

Our goal is to show that ind(B;σ) = 0. We will now prove that ind(B;σ) =ind(Pµ). To do this, we use the spectral flow theorem and a cut-and-paste argu-ment. Let us begin with the application of the spectral flow. Let

V =M ×Rwith the operator

DV ∶= iσ∂t +BLet

S ∶= −2i(DV + i)−1.

Using the same function µ, we get a function µ ∶ V → [0,1]. Consider theoperator

Qµ = 1 + µS.

Lemma 10.12.4. The index of Qµ is equal to ind(B;σ).

Proof. Just calculate

Qµ = 1+µS = 1−2iµ(DV +i)−1 = (DV +i−2iµ))(DV +i)−1 = (DV −i(2µ−1))(DV +i)−1.

Put ψ = 2µ − 1; this is a function as the one used in Theorem 10.11.9. Since theoperator (DV + i)−1 is invertible, we can use Theorem 10.11.9 and conclude that

ind(Qµ) = ind(DV − i(2µ − 1)) = ind(B;σ).

The final argument will be to compare the indices of Qµ and Pµ. Write W =W0 ∪W1, W1 = M × [0,∞). Let H0 = L2(W ) and H1 = L2(V ). Observe thatthe manifolds V and W are the same on the piece W1 (and the operators are thesame one these pieces as well. We claim that the following indices of operators onH0 ⊕H1 are the same:

(10.12.5) ind((Pµ1)) = ind((1

Qµ))

To see this, we introduce a ”grafting operator” gb ∶H0 →H1, for each function bwith support in (0,∞). This is defined by multiplying a function with b, so that ithas support on W1, and then by transplanting this function to the other manifold,namely V . In the same way, we get a grafting operator gb ∶H1 →H0. Observe that


g g = b2

(either way). Slightly more generally, we can take any smooth function b withvalues in the bundle endomorphisms of the bundle E. We will need to know howthe grafting operator relates the two operators R and S.

Lemma 10.12.6. The operators Rgb−gbS and Sgb−gbR are compact (as operatorsH0 →H1, H1 →H0).

Proof. By passing to the adjoints, it is enough to treat one of the two operators.Recall that

Rgb − gbS = −2i((DW + i)−1gb − gb(DV + i)−1)

Since S−1 ∶W(V )→H1 = L2(V ) is bijective and bounded, it is enough to studythe composition (Rgb − gbS)S−1 ∶W(V )→ L2(W ). For u ∈ Γc(V ), we have

gb(DV + i)u + gb′′u = (DW + i)gbu

by the definitions of the operators, here b′′ is the ”function” iσb′. So

(Rgb − gbS)S−1 = RgbS−1 − gb = gb −Rgb′′ − gb = Rgb′′ ∼ 0

by Proposition 10.12.1.

Now we choose two functions a, b on R with a2+b2 = 1, such that supp(b) ⊂ (0,∞)and supp(a) ⊂ (−∞,2). Let

χ = (a −gg a

) .

Using that a2 + b2 = 1, we obtain that χ2 = 1. Therefore

(10.12.7) ind((Pµ1)) = ind(χ(Pµ

1))χ).

But (P ∶= Pµ)

χ(Pµ1)χ = (aPa + g

2 aPg − aggPa − ag gPg + a2) .

The following lemma shows that the index of the right hand side is the index ofQµ. This proves that ind(B;σ) = ind(Qµ) = ind(Pµ) = 0 and finishes the proof ofthe bordism theorem.

Lemma 10.12.8. The operator

(aPa + g2 aPg − ag

gPa − ag gPg + a2) − (1Qµ

)

is compact.

208 JOHANNES EBERT

Proof. We proceed entry by entry. First of all

aPa + g2 − 1 = a2 + aµRa + b2 − 1 = aµRa ∼ 0

since the function aµ has compact support and by Propostion 10.12.1. Next, letus compute

gPa − ag = g(Pa − a) = gµRa = g[µ,R]a + gaµR ∼ 0

because [µ,R] is compact and because aµ has compact support. It remains toshow that gPg + a2 −Q is compact. But

gPg + a2 −Q = g2 + gµRg + a2 − 1 − µS = gµRg − µSTo prove that gµRg − µS is compact (as an operator on H1), we compose with

the invertible S−1 ∶W(V )→ L2(V ). Calculate

(gµRg − µS)S−1 = gµRgS−1 − µ.It is easy to see that gS−1 = R−1g + gb′′ , where b′′ is the ”function” iσb′. So the

above operator is the same as

gµRgS−1 − µ = gµg + gµRgb′′ − µ.The operator gµg−µ is multiplication with a compactly supported function and

hence compact (since we consider it as an operator W → L2, by Rellich). On theother hand, b′′ has compact support, whence Rgb′′ is compact.

10.13. Guide to the literature. There is a wealth of literature on Dirac opera-tors, and we only wish to emphasize [48] and the more recent [54]. Both sourcesalso contain a lot of background material on Clifford algebras (the classical sourcefor which is [6]). I tried to isolate those aspects of the theory that go into the proofof the index theorem, and the exposition differs in many ways from the classicalapproach. The point is to offer an alternative viewpoint.

The classical proof of the K-theoretic version of the index theorem given byAtiyah and Singer [8] used pseudodifferential operators. An exposition of this proofis given in the book by Lawson and Michelsohn [48], and in a more sketchy wayby Shanahan [60]. The proof of the index theorem that we give is due to Guentner[32] and is build on work by Baum-Douglas [12], [13]. The key point is the bordisminvariance of the index, and here we follow Higson [35], with some simplifications.Before they found the proof in [8], Atiyah and Singer gave an alternative argumentusing cobordism theory. This proof is given in the book [55], which contains adifferent proof of the bordism invariance of the index.

There is a completely different version of the proof of the index theorem, basedon the asymptotic expansion of the heat kernel. The first proof along these lineswas given by Atiyah, Bott, Patodi [2]. A textbook reference is [27]; and a moremodern and simpler version was given by Getzler. You can find this proof in thebooks [14] and [56].


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