Proyecto Fin de Master en Investigacion Matematica

Facultad de Ciencias Matematicas

Universidad Complutense de Madrid

Morse theory

Author:

Jose Alberto Iglesias Martınez

Supervisor:

Vicente Munoz Velazquez

2009-2010

ABSTRACT

In Morse theory, one studies the topology of a finite-dimensional smooth manifoldthrough the critical points of a sufficiently well-behaved real valued function on it. Herewe present an introduction to Morse homology from a modern standpoint, with tech-niques similar to the ones used in Floer homology for infinite-dimensional manifolds.

First, we start defining Morse functions, proving existence theorems for them, andconsidering the trajectory spaces of the gradient flow of those functions. We also in-troduce the concept of a Morse-Smale pair, which is the one that is actually useful forMorse homology.

The modern approach of Morse homology mentioned is based on giving said trajec-tory spaces a Banach manifold structure, and through analysis of Fredholm operators,obtain compactifications of the corresponding moduli spaces. Some further work in thisline allows to define a manifold with corners structure on these moduli spaces, and co-herent orientations on their boundaries. This way of tackling the technical difficultiescan be generalized for Floer homology theories, unlike the easier classical dynamicalapproaches to Morse theory.

In a slight detour, we give a brief introduction to sheaf cohomology, prove the deRham theorem, and show how homology can be calculated from currents. While notstrictly needed for the rest, this approach does simplify later work and gives insight intothe different ways in which the same (co)homology arises, for ‘nice’ spaces.

The compactification theorems are then used to define a chain complex formed fromthe critical points of a Morse function, and proving that it’s homology is isomorphicto the singular homology of the underlying manifold, the so-called Morse homologytheorem.

Finally, some applications are presented, and extensions of this theory are hinted at.

Key words: Morse functions, trajectory spaces, Morse homology, sheaf cohomol-ogy, de Rham theorem

MSC2000: primary 58E05; secondary 55N30, 47A13, 37C10, 58A12

RESUMEN

La teorıa de Morse consiste en el estudio de la topologıa de una variedad finito-dimensionala traves de los puntos crıticos de una funcion real definida en dicha variedad, de compor-tamiento suficientemente bueno. Aquı se pretende presentar una introduccion a la mismadesde un punto de vista moderno, con tecnicas similares a las usadas en la homologıaFloer para variedades de dimension infinita.

Primero, se da la definicion de funciones de Morse, teoremas de existencia para lasmismas, y se consideran los espacios de trayectorias del flujo dado por el gradiente dedichas funciones. Tambien se introduce el concepto de par Morse-Smale, el cual es elprincipal para la homologıa de Morse.

El punto de vista moderno mencionado se basa en dar una estructura de variedadBanach a los espacios de trayectorias, y, mediante el analisis de operadores de Fredholm,obtener compactificaciones de los espacios de moduli correspondientes. Con algo masde trabajo se le da a estos ultimos espacios una estructura de variedad con esquinas,y se orientan los bordes correspondientes. Esta forma de trabajar se puede generalizara homologıa Floer, no como los enfoques mas clasicos, aunque sean estos ultimos massencillos.

Dando un cierto rodeo, tambien se presenta una introduccion a la cohomologıa dehaces, se demuestra con ella el teorema de De Rham, y se muestra como es posible calcu-lar homologıa con corrientes. Si bien esto no es necesario para los objetivos principales,simplifica lo que sigue, y ayuda a comprender como la misma homologıa puede surgir dediferentes formas, en espacios suficientemente buenos.

Despues de esto, se usan los teoremas de compactificacion para definir un complejode cadenas desde los puntos puntos crıticos de una funcion de Morse, y probar que lahomologıa de este complejo es isomorfa a la homologıa singular de la variedad sobre laque se trabaja.

Finalmente se presentan algunas aplicaciones, y se dan ideas sobre las generaliza-ciones existentes.

Palabras clave: funciones de Morse, espacios de orbitas, homologıa de Morse,cohomologıa de haces, teorema de De Rham

El/la abajo firmante, matriculado/a en el Master en Investigacion Matematica dela Facultad de Ciencias Matematicas, autoriza a la Universidad Complutense deMadrid (UCM) a difundir y utilizar con fines academicos, no comerciales y mencio-nando expresamente a su autor el presente Trabajo Fin de Master: “Morse theory”,realizado durante el curso academico 2009-2010 bajo la direccion de Vicente MunozVelazquez en el Departamento de Geometrıa y topologıa, y a la Biblioteca de laUCM a depositarlo en el Archivo Institucional E-Prints Complutense con el objetode incrementar la difusion, uso e impacto del trabajo en Internet y garantizar supreservacion y acceso a largo plazo.

Fdo: Jose Alberto Iglesias Martınez Supervisado y autorizado: Vicente Munoz Velazquez

Contents

1 Introduction 21.1 The classical approach . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Modern approaches. Advantages . . . . . . . . . . . . . . . . . . . . . 2

2 Basic definitions and results. Morse functions 4

3 The trajectory spaces 83.1 Trajectory spaces. Banach manifold structure . . . . . . . . . . . . . 83.2 Fredholm operators. Finite-dimensionality . . . . . . . . . . . . . . . 11

3.2.1 Non-trivial bundles . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Transversality. Manifold structure . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Genericity of the Morse-Smale condition . . . . . . . . . . . . 223.4 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 The space of unparametrized trajectories . . . . . . . . . . . . 223.4.2 Compactification result . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Sheaves, cohomology and currents 314.1 Basics. Presheaves and sheaves . . . . . . . . . . . . . . . . . . . . . 314.2 Resolutions of sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 de Rham theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5 Currents and homology . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 The Morse homology theorem 465.1 The Morse chain complex . . . . . . . . . . . . . . . . . . . . . . . . 465.2 The chain homotopy. Morse homology theorem . . . . . . . . . . . . 475.3 The Morse inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Generalizations 536.1 Morse-Bott theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Novikov homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

References 55

1

1 Introduction

In Morse theory, one studies the topology of a finite-dimensional smooth manifoldthrough the critical points of a sufficiently well-behaved real valued function on it.Here we present an introduction to it from a modern standpoint, with techniquessimilar to the ones used in Floer homology for infinite-dimensional manifolds. Some-where in between, with the purpose of presenting different ways in which the usualsingular homology of a manifold arises, we take a detour in order to define sheafcohomology, prove the de Rham theorem and see how homology arises from sheavesof currents. I do not claim originality for any of the results given.

It is quite easy to find motivation for an approach of this kind. Just looking atlevel sets of a height function on a surface, it’s easy to notice that when passinga critical level, the topology of what is left behind changes. For example, in a‘terrain’-like surface S (i.e. the graph of a function f), one would consider the sets

Sa = x ∈ S | f(x) ≤ a.

Now, topologically, going through a valley (minimum) is equivalent to adjoining adisjoint disk to Sa, passing a peak (maximum) is the same as gluing a disk to Sa

along their boundaries, and for a saddle point one attaches a handle.

1.1 The classical approach

The classical approach to Morse theory, as illustrated in the excellent text [Mi],consists in treating the matter quite directly, with rather elementary methods.

The first step is to restrict the investigations to the most adequate kind of func-tions, i.e., the ones whose critical points are all isolated. It’s not hard to prove thatthese always exist using Sard’s theorem.

Then, one tries to reduce the variety of cases to treat, realizing that for anycritical point, only it’s index will be relevant (the so-called Morse lemma). In moreprecise terms, it is possible to find a chart in a neighborhood of a critical point, sothat the function near there is in a standard form only depending on the index ofsaid critical point.

With that, by exactly the same method used in the example above, it is quitestraightforward to prove that any compact manifold has the homotopy type of aCW-complex. This is already an interesting result.

For more delicate investigations into the behavior of Morse functions, the nextstep is to consider their gradient vector field and corresponding flow, in order to knowhow their values increase or decrease. Treating this flow like a dynamical system,and applying a number of results from dynamical systems theory, it is eventuallypossible to prove the Morse homology theorem: homology of the manifold can becalculated as homology of a chain complex formed from the critical points of a Morsefunction, with boundaries arising from the flow lines connecting critical points.

1.2 Modern approaches. Advantages

But the above method is not the only one to get insight about the behavior of Morsefunctions and prove the Morse homology theorem. A more modern approach than

2

the above, and the one we try to introduce in this paper, is to use analytical methodsto analyze the trajectories of the gradient flow.

The starting point is the same, Morse functions and their gradient flows, andone defines the sets

M(p, q)

of flow lines connecting two critical points (it is straightforward to see that for acompact manifolds a flow line must always start and end on critical points). These,as before, arise as solutions to the differential equation

γ = −∇f γ

But instead of a dynamical approach to this equation and its flow, like before,the idea is to apply analytical methods to get properties of the solution spaces, toeventually get enough information about these trajectory spaces so as to get thesame Morse homology theorem. That will be our purpose.

This approach could seem heavy-handed, since just defining appropiate spacesfor these trajectories means getting into Banach manifold territory, and quite someanalysis is needed before everything comes together. The reason for approachingthe theory in this way is mostly pedadogical: while harder and more elaborate, itgives deeper insight about the trajectory spaces, and it more or less easily gener-alizes into other settings, like that of infinite-dimensional starting manifolds (Floerhomologies).

3

2 Basic definitions and results. Morse functions

Definition 2.1. Let M be a smooth manifold, and f : M → R a smooth function.

(i) A point p ∈M is a critical point for f if dpf = 0. A point which is not criticalis called a regular point. Images of critical and regular points through f arecalled critical and regular values, respectively.

(ii) The Hessian of f at p, Hp(f) : TpM → T ∗pM is defined by Hp(f)(v) = ∇v(df),for an arbitrary connection ∇. df vanishing at p guarantees that this doesn’tdepend on the choice of connection, since any two connections differ by atensor.

(iii) A critical point is called nondegenerate, if the Hessian there has null kernel.

(iv) The index of a non-degenerate critical point is defined as the number of neg-ative eigenvalues of the Hessian.

Remark 2.2. One can define the Hessian without resorting to connections. Thisis done by identifying the Hessian with a tensor, Hp(f) : TpM × TpM → R, anddefining:

Hp(f)(v, w) = (V · (W · f))(p) = Vp · (W · f)

For vector fields V , W defined on an open neighborhood of p, such that Vp = v andWp = w. It is easy to check that this doesn’t depend on the extensions used, and infact is equivalent to the previous definition.

Remark 2.3. From this last characterization of the Hessian it is also easy to theck,that if φ : U → Rm is a chart, with U an open neighborhood of p, and φ(p) = 0,The matrix of Hp(f) with respect to the basis ∂

∂x1, . . . , ∂

∂xmhas the following ‘second

derivative’ expression:

Mp(f) =

(∂2(f φ−1)

∂xi∂xjφ(p)

)ij

Definition 2.4 (Morse function). We say a real valued function on a smooth man-ifold is a Morse function, if all of its critical points are nondegenerate.

From these definitions we can get some easy consequences directly:

Lemma 2.5. Non-degenerate critical points are isolated. In particular, a Morsefunction on a compact manifold has a finite number of critical points.

Proof. Consider a chart φ around p as in the above remark. Define the map g :φ(U)→ Rm given by:

g(x) =

(∂(f φ−1)

∂x1

(x), . . . ,∂(f φ−1)

∂xm(x)

)Then g(0) = 0, and d0g = Mp(f) is nonsingular. By the inverse function theorem,g is a diffeomorphism of some neighborhood V of 0, in particular injective in such aneighborhood. This means g(x) 6= 0 for 0 6= x ∈ V , and x is not a critical point forf .

4

The starting point for classical Morse theory is the following lemma, for whichwe don’t give a proof since it won’t be needed for our discussions (one can find aproof in [Mi], for example). However, it does provide some geometrical intuitionabout the behaviour of functions near a non-degenerate critical point.

Lemma 2.6 (Morse lemma). Let p ∈M be a non-degenerate critical point of indexk of a smooth function f : M → R. Then there exists a chart φ : U → Rm, with Uan open neighborhood of p, and φ(p) = 0, such that, in local coordinates:

(f φ−1)(x1, . . . , xm) = f(p)− x21 − . . .− x2

k + x2k+1 + . . .+ x2

m

We will want to study not only critical points but relations between them, seeinghow the values of f increase or decrease. In euclidean space, the obvious choice wouldbe to use the gradient vector field of f . On an arbitrary manifold, we need to get avector field out of df . This is archieved through a metric on the manifold.

Definition 2.7 (Gradient). Let M be a smooth manifold, g a metric on M andf : M → R a smooth function. The gradient of f with respect to g is definedas the image of df through the canonical isomorphism g : TM → T ∗M , given byg(v)(w) = g(v, w). We denote the gradient of f on (M, g) by ∇f .

Our goal, very roughly speaking, will be to understand the behavior of points ofM in the dynamical system defined by the gradient.

Remark 2.8. Throughout this paper we will use the negative gradient −∇f , as iscustomary in the literature.

Proposition 2.9. Let (M, g) be a Riemannian manifold, f : M → R a smoothfunction on f , and ϕt : M → M the local 1-parameter group of diffeomorphismsgenerated by −∇f . For an arbitrary x ∈ M , denote by γx : (a, b)→ M the integralcurve given by γx(t) = ϕt(x). Then limt→+∞ γx(t) and limt→−∞ γx(t) both exist andare critical points for f .

Proof. Let x ∈ M and γx(t) the corresponding flow line of the negative gradient.Since M is compact, γx(t) is defined for all r ∈ R, and the image of f γx is boundedin R. Then, since f decreases along flow lines (very easy to see), we must have:

limt→±∞

d

dt(f(γx(t))) = 0

Let tn be a sequence of real numbers such that tn → −∞. Then γx(tn) has anaccumulation point q since M is compact. Now q must be a critical point of f bythe preceding limit. Pick a neighborhood U of q in which q is the only critical point.Now assume limt→−∞ γx(t) 6= q. Then there is another sequence sn and anotherneighboorhood V ⊂ U of q such that sn → −∞, but γx(sn) ∈ U − V . Then, asabove, γx(sn) must have an accumulation point, which must be a critical point off . But this contradicts that q is the only critical point in U .

Definition 2.10 (Stable and unstable manifolds). Let p ∈ M be a non-degeneratecritical point of f.

5

(1) The unstable manifold of p is defined to be:

W u(p) =

x ∈M | lim

t→−∞γx(t) = p

(2) The stable manifold of p is defined to be:

W s(p) =

x ∈M | lim

t→+∞γx(t) = p

That these sets are actually manifolds will be proved in the next section. We

will also call them the descending manifold of p and the ascending manifold of p,for W u(p) and W s(p) respectively.We have defined Morse functions, but so far we don’t even have an existence resultfor them. We obtain such a result now. The proof is based on the following well-known theorem, for which a proof can be found in [GuiPo]

Theorem 2.11 (Sard’s theorem). Let f : M → N be a smooth map between mani-folds M and N , and C the set of critical points of f . Then f(C) has measure zeroin N.

Let’s first prove our theorem in an euclidean setting (i.e., locally):

Lemma 2.12. Let f : U → R be a smooth function defined on an open set U o fRm. Then for almost all a = (a1, . . . , am) ∈ Rm, the function defined by fa(x) =f(x)−

∑mi=1 aixi is a Morse function.

Proof. Consider the function g given by:

g(x) =

(∂f

∂x1

(x), . . . ,∂f

∂xm(x)

)And let a be a regular value of g. By Sard’s theorem, this is true for almost all a.Now let fa be as in the statement, and let p ∈ Rm be a critical point of fa. Then:

dpfa = g(p)− a = 0

Which means that g(p) = a, a regular value. Then dpg is surjective, and henceinvertible. But the Hessian Hp(fa) is precisely dpg. So p is nondegenerate.

For our result, we need another of the basic theorems in differential topology. Aproof can be found in [Br].

Theorem 2.13 (Whitney embedding theorem). If M is a smooth manifold of di-mension n, there exists a smooth embedding g : M → R2n+1

So we can assume our manifold is embedded in euclidean space, which enablesus to use the lemma.

Theorem 2.14 (Existence of Morse functions). Let M be a smooth manifold em-bedded in Rr. For almost all a = (a1, . . . , ar) ∈ Rr, the function f : M → R givenby f(x) =

∑rj=1 ajxj is a Morse function.

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Proof. Let x = (x1, . . . , xr) ∈ M ⊂ Rr. Then, there is a neighborhood U of x inwhich a subset of the coordinates, say xj1 , . . . xjm form local coordinates for M . Tosee this, one just needs to note that since TxM → TxRr is injective, it’s dual T ∗xRr →T ∗xM is surjective. Hence, for some neighborhood U of x, T ∗xM is generated by alinearly independent set dxj1 , . . . , dxjm . Then xj1 , . . . , xjm are linearly independentand hence form a coordinate system on U.Now cover M by open sets Uj of this kind, and let (x1, . . . , xm) be a coordinatesystem on Uj (assuming, without loss of generality, jk = k). Choose (am+1, . . . , ar) ∈Rr−m, and consider the function f(x) =

∑rj=m+1 ajxj. By the lemma, the function:

fa(x) = f(x)−r∑j=1

ajxj

is a Morse function for almost all a = (a1, . . . , am), fa. Hence, for almost all(a1, . . . , ar),

∑rj=1 ajxj is Morse on Uj. Let Aj be the points a ∈ Rr for which

this last function is not Morse on Uj. Each of these has measure zero, and henceA =

⋃j Aj also has measure zero.

Remark 2.15. In fact, a number of stronger statements than the ones above canbe proved, such that the set of Morse functions is generic among smooth functionson M (in a Baire category sense), and that there are Morse functions arbitrarilyuniformly close to a given smooth function.

Actually, one needs to require a bit more from our functions for them to beuseful in homology calculations, namely, transversality of intersections of stable andunstable mainfolds.

Definition 2.16 (Morse-Smale function). Given a Riemannian manifold (M, g), asmooth function f : M → R is a Morse-Smale function if for all p, q ∈ Crit(f),W u(q) t W s(p), where Crit(f) denotes the set of critical points of f .

Remark 2.17. Note that this transversality condition depends on the metric g, sinceit is defined through the negative gradient field of f . Hence, since we want to provetopological results, it suffices to find a metric in which the above condition holds. Itwill be proved in the next section that this is the case for a large class of Riemannianmetrics on M , for an arbitrary Morse function.

7

3 The trajectory spaces

We start by presenting a theorem that is more powerful than we need, and conse-quently more difficult than we will be able to prove. The objective of this full sectionwill be to prove, with varying levels of detail, the part of it that is actually useful forthe Morse homology theorem, that is, finiteness of the number of trajectories whenthe relative index is one, and coherent orientations for that case.

Theorem 3.1. If M is a closed smooth manifold, and (f, g) is a Morse-Smale pairon M , then for any two critical points p, q ∈ M , the trajectory space of flow linesconnecting them, M(p, q) has a natural compactification to a smooth manifold withcorners M(p, q), whose codimension k stratum is

M(p, q)k =⋃

r1,...,rk∈Crit(f)

M(p, r1)×M(r1, r2)× . . .×M(rk, q)

with r1, . . . , rk, q all different. For the case k = 1, the boundary of M(p, q), asoriented manifolds, we have:

∂M(p, q) =⋃

r∈Crit(f)

(−1)ind p+ind r+1M(p, r)×M(r, q)

Some of these terms haven’t even been defined yet, but the idea is clear. Whatone wants to show, is that the trajectory spaces associated to the gradient flow off are compact up to broken trajectories, of which each part is a flow curve itself.Moreover, one can give orientations to these boundaries in a coherent way.

To do this, first one has to define adequate topologies and structure on curvespaces, which in this case will turn out to be that of infinite-dimensional Banachmanifolds. The next step is, through analysis of certain Fredholm operators, toobtain a finite-dimensional manifold structure on the trajectory spaces themselves.After that, one can work towards the compactness result itself, again with suitablemodes of convergence for the trajectory spaces. Then, it will be possible to prove‘gluing’ results to perturb broken flow lines to actual flow lines, in such a way thatthe process can be parametrized and forced to converge to the original broken flowline, so as to obtain the mentioned manifold-with-corners structure. The final laststep will be to obtain coherent orientations for all these spaces.

This whole section follows the work done in [Sch].

3.1 Trajectory spaces. Banach manifold structure

In this section we define the trajectory spaces we need to work with. To mimic thebehavior of flow lines, we start compactifying R in the following way:

Definition 3.2. Let R = R ∪ ±∞, equipped with the structure of a boundedmanifold, by the requirement that:

h : R→ [−1, 1]

t→ t√1 + t2

8

be a diffeomorphism. Additionally, for x, y ∈ M we define the set of smooth,compact curves C∞x,y by:

C∞x,y = C∞x,y(R,M) = u ∈ C∞(R,M) | u(−∞) = x, u(+∞) = y

Directly from this definition, we can get a characterization of the asymptoticdecrease of C∞(R) functions imposed by the differentiable structure of R.

Lemma 3.3. For each f ∈ C1(R,R) there is a constant c(f) > 0, such that thefollowing estimate holds:

|f ′(t)| ≤ c(f)

(1 + t2)32

Actually, the following useful estimates are also true:

Corollary 3.4. Given A ∈ C1(R,GL(n,R)), there is a constant c(A) > 0 such thatthe estimate

‖As‖1,2 ≤ c(A)‖s‖1,2

holds for all s ∈ H1,2(R,Rn), and where (As)(t) = A(t) · s(t), t ∈ R.

Corollary 3.5. Let f ∈ C1(R,R) satisfy the condition f(±∞) = 0. Then f ∈H1,2(R,R).

Definition 3.6. Now let ξ ∈ VecR be a C∞(R)-smooth, finite-dimensional vector

bundle on R, and let φ : ξ∼=→ R × Rn be a smooth trivialization. Then using the

induced one-to-one mapping φ∗ between the associated vector spaces of sections, wecan define:

H1,2R (ξ) = φ−1

∗ (H1,2(R,Rn)) = φ−1∗ (s) | s ∈ H1,2(R,Rn)

Now, the first of the corollaries above implies that φ∗ induces a banach space topol-ogy on the vector space H1,2

R (ξ) in a way which is independent of the particular choiceof trivialization φ, since the change from one trivialization to another is representedby some A ∈ C∞(R,GL(n,R)).

It is convenient to note that this space hasn’t been constructed from a measureon R, but from isomorphisms with H1,2(R,Rn).

Now, consider the exponential map on the complete Riemannian manifold M ,

exp : TM ⊃ D →M

where D is a convex neighborhood of the zero section where the exponential isdefined everywhere. We denote by h∗D the induced open and convex neighborhoodof the zero section in the pullback bundle h∗TM , for a smooth, compact curveh ∈ C∞(R,M).

Definition 3.7. From the Sobolev embedding H1,2loc → C0, we can define

exph : H1,2R (h∗D)→ C∗(R,M)

s→ exp s

where (exp s) = exph(t) ·s(t), which is well defined for h ∈ C∞(R,M). Thus we candefine:

P1,2x,y = P1,2

x,y(R,M) = exp s ∈ C0(R,M) | s ∈ H1,2R (h∗D), h ∈ C∞x,y(R,M)

9

A proof for the three key propositions below can be found in [Sch], appendix 1:

Proposition 3.8. The set of curves P1,2x,y ⊂ C0

x,y(R,M) with given endpoints isequipped with a Banach manifold structure via the atlas of charts

H1,2R (h∗D), exphh∈C∞x,y(R,M)

Additionally, the following inclusions hold:

C∞x,y(R,M)dense⊂ P1,2

x,y(R,M)dense⊂ C0

x,y(R,M)

Moreover, there is a countable sub-atlas.

We shall use the following representation of the tangent space

TP1,2x,y = H1,2

R (P1,2∗x,y TM) =

⋃x∈P1,2

x,y

H1,2R (s∗TM)

Now this is a Banach bundle on P1,2x,y with H1,2(R,Rn) as characteristic fiber. By

analogy to H1,2R we can define a section functor

L2R : VecC∞(R)→ Ban

which is endowed with a Banach space topology given by L2(R,Rn). This directsus to the Banach bundle

L2R(P1,2∗

x,y TM) =⋃

s∈P1,2x,y

L2R(s∗TM)

Taking the above bundle structure into account, the second key proposition is:

Proposition 3.9. Let f ∈ C∞(M,R) be an arbitrary smooth real function on M .Then, given critical points x, y ∈ Crit(f) as endpoints, the gradient field ∇f inducesa smooth section in the L2-Banach bundle,

F : P1,2x,y → L2

R(P1,2∗x,y TM)

s→ s+∇f s

Actually, the study of the trajectory spaces is founded upon exactly this sectionF , as shown by the following proposition:

Proposition 3.10. The zeroes of the section F : P1,2x,y → L2

R(P1,2∗x,y TM) are exactly

the smooth curves which solve the differential equation s = −s + ∇f s, and alsosatisfy the conditions limt→−∞ s(t) = x and limt→+∞ s(t) = y.

Definition 3.11 (Trajectory spaces). We denote the above trajectory spaces ofsolutions to

s = −s+∇f s,

subject to limt→−∞ s(t) = x and limt→+∞ s(t) = y, by M(x, y).

10

We also have the following useful lemma, which can be proved in an elementarybut somewhat involved way:

Lemma 3.12. Let X : U(0) → Rn be a C1 vectori field defined on a neghborhoodof 0 ∈ R, and let 0 be a critical point of X such that the linearization DX(0) isnon-degenerate and symmetric. Then there is an ε > 0 such that the solutions to

s = X(s) with limt→∞

s(t) = 0

satisfy the following estimate: There are constants c > 0 and t0 ∈ R depending ons, satisfying

|s(t)| ≤ ce−st for all t ≥ t0

Note that this will be the case for the gradient vector field and correspondingflows, so we have some estimate about the asymptotic behaviour of curves in thetrajectory spacces.

3.2 Fredholm operators. Finite-dimensionality

The next step consists in showing that the above operator F is Fredholm. First,we will prove the analogous result considering linear operators on the trivial bundleR× Rn, of the type

(FAs)(t) = s(t) + A(t) · s(t)with s ∈ H1,2(R,Rn) and A ∈ C0

b (R,End(Rn)). Let’s settle some notation first.We say an operator A ∈ End(Rn) is conjugated self-adjoint if it is self-adjoint withrespect to some scalar product. Also, we will denote:

X = H1,2(R,Rn), Y = L2(R,Rn)

S = A ∈ GL(n,R) | A is conjugated self-adjointA = A ∈ C0(R,End(Rn)) | A± = A(±∞) ∈ S

The set S is the set of Hessians appearing in Morse theory. So we can define:

µ(A) = #(σ(A) ∩ R−) for A ∈ S

to be the so-called Morse index of A, where σ(A) is the spectrum of A. We can alsoregard A as a normed space, with respect to ‖ · ‖∞.

We wish to study the following map:

C0b (R,End(Rn)) 3 A→ (FA : X → Y )

Where (Fas)(t) = s(t) + A(t) · s(t). We denote this map by F . This is an affinemap, continuous by the estimate:

‖(FA − FB)(s)‖0 =

(∫R|(A−B)s|2dt

) 12

≤ ‖A−B‖∞‖s‖0 ≤ ‖A−B‖∞‖s‖1

Hence ‖FA − FB‖L(X,Y ) ≤ ‖A − B‖∞. The central result of this section is thefollowing

11

Proposition 3.13. For all A ∈ A, the linear operator FA : X → Y is Fredholm.

To prove it, we will need the following lemma (a semi-Fredholm operator isone with either finite dimensional kernel, or finite dimensional cokernel, but notnecessarily both):

Lemma 3.14. Let X, Y, Z be Banach spaces and F ∈ L(X, Y ), K ∈ K(X,Z) andc > 0 with

‖x‖X ≤ c(‖Fx‖Y + ‖Kx‖Z), for all x ∈ X

Then F is a semi-Fredholm operator.

Proof. Define the setS = x ∈ kerF | ‖x‖X = 1

and consider a sequence (xk) ⊂ S. Then, by our assumption, and the identityFxk = 0 for all k ∈ N we get

‖Xk − xl‖X ≤ c‖K(xk − xl)‖Z , for allk, l ∈ N

Given that K is compact, the sequence (Kxk), and therefore (xk) have convergentsubsequences. Thus S is also compact, and so kerF is of finite dimension.

Now, from the Hahn-Banach theorem and this finite dimension, there exists aclosed subspace X0 ⊂ X satisfying

kerF ⊕X0 = X

Consider now a sequence (Fxk) ⊂ R(F ) converging to Y , so without loss of gener-ality we can assume

(xk)k∈N ⊂ X0, Fxk → y ∈ Y

Assume (xk) is unbounded. Then we switch to the sequence xk‖xk‖

. Thus we can

assume (maybe passing to a subsequence) that

(xk)k∈N, ‖xk‖X = 1, ‖Fxk‖Y ≤1

kfor all k ∈ N

Using the hypothesis again, there is a convergent subsequence of (xk), since (Fxk)converges and K is compact. Thus, denoting the subsequence again by (xk),

xk → x, ‖x‖X = 1 and Fx = 0

in contradiction to the construction of X0. Therefore, the sequence (xk) must bebounded, so that the same argument yields a convergent subsequence. This impliesthe identity

y = Fx, for some x ∈ X0

hence R(F ) is closed in Y .

Proof of the proposition. It will consist in four parts, formulated as lemmas below.

12

Lemma 3.15 (Part 1). Let A ∈ A be a constant map from R to S. Then there isa constant c > 0, such that the estimate

‖s‖1 ≤ c‖FAs‖0 holds for all s ∈ X

Proof. By F : Y → Y we denote the Fourier isometry, where we use the notationsX = H1,2(R,Cn) and Y = L2(R,Cn) throught the proof. In particular,

F(S)(t) = itF(s)(t), t ∈ R for all s ∈ X

Also, let ω : F(X) → Y be the operator ω(s)(t) = t · s(t), t ∈ R so that we obtainthe identity

F(FA(s)) = (iω + A) F(s)

and thereforeFa = F−1 (iω + A) F : X → Y (3.1)

So let us assume A ∈ S, so that λ0 = min |σ(A)| > 0 holds. We consider

B1(ω), B2(ω) : Cn → Cn,

B1(ω) · x = (1 + ω2)12x, ω ∈ R,

B2(ω) = iω + A, ω ∈ R

and hence0 /∈ σ(B2(ω)) = iω + σ(A)

This implies that the inverse B2(ω)−1 exists and stisfies

‖B−12 (ω)‖ = sup

λ∈σ(B2(ω))

|λ−1| = supλ∈σ(A)+iω

1

|λ|=

1√λ2

0 + ω2

Summing we obtain

‖B1(ω) B−12 (ω)‖ ≤

√1 + ω2

λ20 + ω2

≤ max(1

λ0

, 1) = c(A)

So we regard B1, B−12 ∈ C0(R,End(Cn)) as multiplication operators in L(X, Y ),

obtaining the inequality‖B1B

−12 ‖ ≤ c(A) (3.2)

Putting all this together we get the needed estimation, combining

‖s‖21 = ‖

√1 + ω2Fs‖2

0 = ‖F−1B1B−12 FF−1B2Fs‖2

0

with〈F−1B1B2Fξ, η〉 0 = 〈B1B

−12 Fξ,Fη〉0 ≤ ‖B1B

−12 ‖∞‖ξ‖0‖η‖0

By the inequality (3.2) this yields

‖F−1B1B−12 F‖L(X,Y ) ≤ c(A),

so that we conclude the estimate

‖s‖1 ≤ c(A)‖F−1B2Fs‖0

and taking (3.1) into account we get the inequality of the lemma statement.

13

Lemma 3.16 (Part 2). Given any A ∈ A, there are constants T > 0, c(T ) > 0,that satisfy

‖s‖1 ≤ c(T )‖FAs‖0 for all s ∈ X, s|[−T,T ] = 0

Proof. From step 1 we have

‖s‖1 ≤ c(A±)‖FA±s‖0 for all s ∈ X

We therefore define c = max(c(A+), c(A−)). Given ε > 0and A ∈ A, there is aTε > 0 large enough such that

‖A−A(t)‖ ≤ ε for all t ≤ −Tε

‖A+ − A(t)‖ ≤ ε for all t ≥ Tε

Restricting ourselves to s± ∈ X with s|−[−Tε,∞) = 0 and s|+(−∞,Tε] = 0, we get

‖FA±s±‖0 ≤ ‖FAs±‖0 + ‖(FA± − FA)s±‖0 ≤ ‖FAs±‖0 + ε‖s±‖0

Now consider s ∈ X such that s|[−Tε,Tε] = 0. Then we find s± as above, fulfillings = s+s− such that

‖s‖1 = ‖s−‖1 + ‖s+‖1 ≤ c(‖FA−s−‖0 + ‖FA+s+‖0)

≤ c(‖FAs−‖0 + ‖FAs+‖0) + cε(‖s−‖0 + ‖s+‖0)

= c‖FAs‖0 + cε‖s‖0

And this finally implies‖s‖1 ≤ c‖FAs‖0 + cε‖s‖1

and for an ε < 1/c and appropiate T (ε)

‖s‖1 ≤1

1− cε‖FAs‖0, for all s ∈ X with s|[−T (ε),T (ε)] = 0

Lemma 3.17 (Part 3). For any A ∈ A, there is Banach space Z and a K ∈K(X,Z), c > 0 satisfying

‖x‖X ≤ c(‖FA‖Y + ‖Kx‖Z) for all x ∈ X

In fact, FA is a semi-Fredholm operator.

Proof. Let T be the constant obtained in step 2. Then∫ T

−T|s+ As|2dt =

∫ T

−T

(|s|2 + 2〈s, As〉+ ‖As‖2

)dt

=

∫ T

−T

(1

2|s|2 − |As|2

)dt

14

due to the computation

|s|2 + 2〈s, As〉+ |As|2 ≥ 1

2|s|2 − |As|2

from |s + 2As|2 ≥ 0. Therefore, using |A(t) · s(t)| ≤ ‖A(t)‖ · |s(t)| and settingc = max[−T,T ] ‖A(t)‖, we conclude∫ T

−T|s+ As|2dt ≥ 1

2

∫ T

−T|s|2 − c

∫ T

−T|s|2dt

Hence, there is a c > 0 satisfying∫ T

−T

(|s|2 + |s|2

)dt ≤ c

∫ T

−T

(|s|2 + |s+ As|2

)dt. (3.3)

Now, defining a cut-off function β ∈ C∞(R, [0, 1]) with the properties

β(t) =

0, |t| ≥ T + 1

1, |t| ≤ Tand β(t) 6= 0 for |t| ∈ (T, T + 1)

we can combine the estimate (3.3) with step 2 and get

‖s‖1 = ‖βs+ (1− β)s‖1 ≤ ‖βs‖1 + ‖(1− β)s‖1

≤ c(‖βs‖0 + ‖FA(βs)‖0 + ‖FA((1− β)s)‖0)

for c > 0 big enough. That is

‖s‖1 ≤ c(‖βs‖0 + 2‖βs‖0 + ‖βFAs‖0 + ‖(1− β)FAs‖0

)≤ c

(‖s‖L2([−T−1,T+1]) + ‖FAs‖0

)And considerng the following composition of a continous restruction map and theRellich-Kondrachov compact embedding:

K : H1,2(R,Rn)→ H1,2([−T − 1, T + 1],Rn) → L2([−T − 1, T + 1],Rn) = Z

we obtain

K : X → Z

s→ s|[−T−1,T+1] ∈ L2([−T − 1, T + 1])

as a compact operator. Thus lemma (3.14) completes the proof

Lemma 3.18 (Part 4). FA is a Fredholm operator.

Proof. Up to now, we know that FA has finite-dimensional kernel, and that it’sarange is closed in Y . So the cokernel of FA is a banach space satisfying cokerFA ∼=R(FA)⊥, because Y is a Hilbert space. Thus, let r ∈ R(FA)⊥, that is

〈r, s+ As〉0 = 0for any s ∈ H1,2.

15

In particular, we can deduce

〈r, ω〉0 = −〈Atr, φ〉0, for all φ ∈ C∞0 (R,Rn)

But, by definition, this means r is weakly differentiable, with r = At · r ∈ L2. Hencer ∈ W 1,2.

Then we know that r ∈ X and r ∈ kerF−At , beacuse −At ∈ A. Therefore, thereis an isomorphism

cokerFA ∼= kerF−At

And working analogously to the proof sof step 3, we have dim cokerF−At <∞. Thisconcludes the proof.

Now our aim is to express the associated Fredholm index in terms of the Morseindices of the critical points. At this point, only the changes of sign of eigenvaluesof A ∈ A is important. The proof will consist in transforming A in such a way thatthe Fredholm index doesn’t change, but can be easily analyzed.

Definition 3.19. From the proposition, we consider the subset

Σ = F (A) = FA ∈ L(X, Y ) | A ∈ A ⊂ F(X, Y )

and denote the equivalence class of operators from Σ with respect to the relationB± = A± by

ΘFA = FB ∈ Σ | B± = A±, A ∈ A

The folowing lemma will be crucial:

Lemma 3.20. Given F ∈ Σ, the class ΘF is contractible within Σ as a subspace ofF(X, Y ).

Proof. Let F = FA0 ∈ Σ be arbitrary and define Θ = ΘF . Then we study the map(just like a linear homotopy):

κ : [0, 1]×Θ→ Θ with κ(τ, FA) = FA(τ)

A(τ) = (1− τ) · A+ τ · A0, that is

FA(τ) ∈ Σ for all τ ∈ [0, 1], since A(τ)± = A± = A±0

It is clear thatκ(0, ·) = idΘ and κ(1, ·) = AAΘ

Thus, we must show the continuity of κ in both variables, so let us start with

limn→∞

τn = τ and limn→∞

FAn = FA (3.4)

that islimn→∞

‖FAn − FA‖L(X,Y )=0.

so we must check that

limn→∞

‖FA(τ) − FAn(τn)‖L(X,Y )=0.

16

So, let ε > 0 and nk a subsequence (nk)k∈N satisfying

‖FA(τ) − FAnk (τnk )‖L(X,Y ) ≥ ε

Taking into account equation (3.4) above,

‖(A(τ)− Ank(τnk)) · unk‖0

=‖[(1− τ) · A+ τA0 − (1− τnk) · Ank − τnkA0] · unk‖0

=‖(A− Ank) · unk + (τ − τnk)A0 · unk+ (τnkA− τA+ τnkAnk − τnkA) · unk‖0

≤‖FA − Fnk‖L(X,Y ) + |τ − τnk |‖A0‖L(X,Y )

+ |τ − τnk |‖A‖L(X,Y ) + |τnk |‖FA − FAnk‖L(X,Y ) → 0

which is a contradiction.

This last lemma implies that the index map ind : Σ→ Z must be constant whenrestricted to one of the ΘFA , and this means that indFA is determined uniquely bythe endpoints A± ∈ S. Knowing this, we carry out an appropiate conjugation ofFA:

Lemma 3.21. For any A ∈ A, we can find a D ∈ A of the form diag(λ±1 , . . . , λ±n )

withindFD = indFA

Proof. Since A ∈ A we have A± ∈ S. Then there is C± ∈ GL(n,R) such that

C±A±(C±)−1 = diag(λ±1 , . . . , λ±n )

so that the eigenvalues are ordered by sign, that is, signλ±i ≥ signλ±i+1.Also, the ends C± can be chosen in order to satisfy detC± > 0. so that both

C+ and C− lie in the same pathwise connected component of GL(n,R). Thus thereis a curve

C ∈ C∞(R,GL(n,R)) with ends C(±∞) = C±

which is also eventually constant. This means

C(t) =

C+, t ≥ T

C−, t ≤ −Tfor some T > 0

We shall henceforth denote by C both multiplication operators

CX : X → X and CY : Y → Y

s→ C · sIt’s obvious that C represents a linear isomorphism and that the following identitieshold:

(CY FAC−1X )(s)(t) = C(t) · ( ∂

∂t+ A(t)) · (C−1(t) · s(t))

= s(t) +

(C(t) · ∂

∂t(C−1)(t) + C(t)A(t)C−1(t)

)· s(t)

=(FC ∂

∂t+CAC−1s

)(t)

17

Here, ∂∂t

(C−1)(t) = 0 holds for |t| ≥ T . Thus we compute the ends as(C∂

∂t(C−1) + CAC−1

)±= diag(λ±1 , . . . , λ

±n ) = D±

obtaining the relationCY FAC

−1X ∈ ΘD±

Now, given that CX : X → X and CY : Y → Y are isomorphisms, the identity

ind(CY FAC−1X ) = indFA

Follow from the composition rule for Fredholm indices.

And we finish with:

Proposition 3.22. Given any A ∈ A, the Fredholm index of FA equals the relativeMorse index,

indFA = µ(A−)− µ(A+)

Proof. It is sufficient to compute kerFA and cokerFA when A is of the shape:

A(t) = diag(λ1(t), . . . , λn(t)), and

signλi(±∞) ≥ signλi+1(±∞), i = 1, . . . , n− 1

and λi(t) constant for |t| ≥ 1

Now, s ∈ H1,2(R,Rn), s(t) = (s1(t), . . . , sn(t)) is in kerFA if and only if it representsa global solution of the sysem of differential equations:

st = −λi(t) · si(t), i = 1, . . . , n

with the bounded functions λi : R → R. Now, by explicit calculation, for largetimes we have

si(t) = −λi(t) · si(t), t ∈ R and 0 6= si ∈ H1,2(R,R)

if and only if

λ−i < 0 and λ+i > 0, as si(t) =

e−λ

−i t, t < −1

e−λ+i t, > 1

And, as the eigenvalues are ordered by sign, we compute

dim kerFA = #k ∈ 1, . . . , n | λ−k < 0 and λ+k > 0

= max(µ(A−)− µ(A+), 0)

Analogously we compute

dim cokerFA = max(µ(−A−)− µ(−A+), 0) = max(µ(A+)− µ(A−), 0)

from the isomorphism cokerFA ∼= kerF−At ∼= kerF−A. Now, the difference betweenthe two dimensiones above gives indFA = µ(A−)− µ(A+).

18

3.2.1 Non-trivial bundles

The case that is actually useful for Morse theory is that of a smooth vector bundle ξon R endowed with a Riemannian metric, so that H1,2(ξ) and L2(ξ) are well defined,as in the previous section. The principal obstruction is that one has to generalizethe time derivation to some covariant derivation that might disturb the Fredholmproperty.

The key to all this turns out to lie in that the Christoffel symbols associatedto the connection and any trivialization should vanish asymptotically (i.e, at ±∞).If this happens, the covariant derivation is called Fredholm-admissible, and thecorresponding operators are, as expected, Fredholm. It is obvious that the bundlesand connections arising in our context satisfy this condition. More details can befound in [Sch].

3.3 Transversality. Manifold structure

We now focus on endowing the trajectory spaces a finite-dimensional manifold struc-ture, building upon the results of the previous one together with the Morse-Smalecondition. The result we are aiming for is the following:

Theorem 3.23. If (f,g) satisfy the Morse-Smale condition, the trajectory spacesM(x, y) are closed submanifolds of P1,2

x,y of finite dimension µ(x)− µ(y).

The proof of this result will be quite technical, so we need to refresh somedefinitions first.

Definition 3.24 (Genericity). In a Baire space X, we call a subset Σ ⊂ X a G-set,if it is a countable intersection of open and dense subsets. A subset G ⊂ X is genericwith respect to some condition if the condition holds for a G-set Σ ⊂ G.

It is now time to enunciate the above theorem in a more general (and technical)fashion, which hints at the way it’s proven:

Proposition 3.25. Let G and M be Banach manifolds and τ : E → M a Banachbundle on M with fiber E. Additionally, let Φ : G × M → E be a smooth G-parameter section, i.e. Φ smooth and Φ(g, ·) a smooth section in E for each g ∈ G.Furthermore, let Φ satisfy the condition:

There is a countable trivialization (U, ψ), ψ : E|U∼=U ×EU , such that for each

(U, ψ):

(a) 0 is a regular value of pr2 ψ Φ : G× U → EU

(b) pr2 ψ Φg : U → EU is a Fredholm map with index r for al g ∈ G.

Then there is a G-set Σ ⊂ G such that the set

Zg = Φ−1g (0) = m ∈M | Φg(m) = 0

is a closed submanifold of M for al g ∈ Σ.

19

Remark 3.26. Let us translate the result into our starting Morse theory terms:Our G will be a Banach space of metrics to be defined later (not trivial), the G-

parameter section will be the one from the previous sections, that is, for a starting

manifold N , M = P1,2x,y , E = L2(P1,2∗

x,y TN) amd Φg = F where sF→ s + ∇f s is

the one from proposition 3.9, which implicitly depends on the metric through thedefinition of the gradient.

As for the hypotheses (a) and (b), they are part of the Fredholm results inthe previous section, when specialized to this situation, subject to the countablesub-atlas of proposition 3.8.

And the result itself provides us with both the manifold structure of the trajec-tory spaces, and the Morse-Smale condition for a generic set of metrics ⊂ G.

For the proof, we will need the following somewhat-standard Banach spacelemma, which we dont prove here:

Lemma 3.27. Let Φ be a bounded, linear map of Banach spaces, which is onto andof the form

Φ : E × F → G, Φ(e, f) = Φ1(e) + Φ2(f)

and such that Φ1,Φ2 are continuously linear, and let Φ2 : F → G be a Fredholmoperator. Then there is a decomposition

E × F = ker Φ⊕H

such that H is closed in E × F .

We will also need the Sard-Smale theorem, which is a generalization of Sard’stheorem for Fredholm maps on Banach manifolds:

Theorem 3.28. Let f : M → V be a Cq Fredholm map with q > max(ind f, 0).Then the regular values of f is a G-set in V .

Proof. Actually not too complicated, can be found in [Sm].

Proof of proposition 3.25. Owing to the countable trivialization of E we can assumewithout loss of generality that E is a trivial bundle. This is to say that there is asmooth map

Φ : G×M → E

with respect to the Banach space E, such that 0 ∈ E is a regular value and the mapsΦg : M → E are Fredholm maps of index r for all g ∈ G. This assumption can bemade since Σ =

⋂U ΣU is again a G-set if we consider the countable trivialization

(U, ψU). ThusZ = Φ−1(0) ⊂ G×M

is a smooth Banach manifold with associated tangent spaces given by

TzZ = kerDΦ(z) for all z = (g,m) ∈ Z

as follows from local coordinate charts, lemma 3.27 and the implicit function theo-rem.

20

Now we need to conclude that the restriction of the projection map to thisBanach manifold Z,

π : Z → G

is a Fredholm map endowed with the same index as Φg, if we consider z = (g,m) ∈ Z.First, let us focus on the kernels of the mapsDπ(x) : TzZ → E andD2Φ(z) : TmM →E. If we assume D2Φ(z) Fredholm, we obtain the identity (of finite dimensionalvector spaces)

kerDπ(z) = TzZ ∩ TmM = kerD2Φ(z) (3.5)

Next, we consider the following homomorphism between the quotient vector spaces:

D1Φ : TgG/R(Dπ)→ E/R(D2Φ)

[v]R(Dπ) → [D1Φv]R(D2Φ)

having in mind the identities

v = Dπ · (v, w) and D1Φ · v = −D2Φ · w

for each pair (v, w) ∈ TzZ ⊂ TgG × TmM , we deduce the isomorphism property of

D1Φ from DΦ(z) being onto. Now, knowing that cokerD2Φ(z) is of finite dimension,

D1Φ appears to be an isomorphism of finite-dimensional cokernels,

D1Φ : coker(Dπ(z))∼=→ cokerD2Φ(z) (3.6)

Hence, 3.5 and 3.6 imply the Fredholm property of the projection map π. Now wecan apply the Sard-Smale theorem to this map and get a G-set of regular values ofπ.

Now, to see DΦ(z) onto, let b ∈ Σ. If Φ(b,m) does not vanish for any m ∈ M ,0 is trivially a regular value of Φb. Therefore, let us consider an m ∈ M withΦ(b,m) = 0. Then, we have to show the operator

DΦb(m) = D2Φ(b,m) : TmM → E

to be onto. Let γ ∈ E be arbitrary. Due to regularity of b with respecto to Φ, thereis a pair (α, β) ∈ TbG× TmM satisfying

γ = DΦ(b,m) · (α, β) = D1Φ · α +D2Φ · β (3.7)

But b is also regular with respecto to π : Z → G, that is, for any α ∈ TbG one canfind an (α′, β′) ∈ T(b,m)Z such that it holds

α′ = Dπ(b,m) · (α′, β′) = α

Hence (α′, β′) ∈ kerDΦ(b,m), that is

0 = Dφ(b,m) · (α′, β′) = D1Φ · α +D2φ · β′ (3.8)

Substractiong 3.8 from 3.7, we find a preimage of γ under D2Φ(b,m) of the form

γ = D2Φ(b,m) · (β − β′)

and we are finished.

21

3.3.1 Genericity of the Morse-Smale condition

Given a Morse function, there is a set of metrics generic with respect to the Morse-Smale condition. This will be derived from proposition 3.25, where the only thingremaining is to define a suitable Banach manifold of metrics.

We will start with a fixed, arbitrary metric g0 and find a generic set of variationswith respect to g0. By this we mean a set of smooth sections A ∈ End(TM) whichare fiberwise self-adjoint with respect to g0, positive definite and which differ fromthe identity section by a small amount.

Such a set can be explicitly given the structure of a Banach manifold, by defininga certain kind of norms on sections of any vector bundle. The details are rathertechnical, and can be found in [Sch] and [Flo].

3.4 Compactification

In this section, we want to study the compactness properties of the trajectory spaces.The implicit function theorem implies that the finite-dimensional manifoldsM(p, q)are manifolds without boundary, since they arise from the differential equation

γ = −∇f γ

3.4.1 The space of unparametrized trajectories

Before going into the main part of the section, it will be useful to obtain some moreinformation about the trajectory manifold M(x, y). We want to analyze one of it’sessential properties, that is, it’s symmetry with respect to ‘time-shifting’, or moreformally, additive reparametrization of the trajectories.

First, let’s ground some notation. Let γ : R→M be a solution of the differentialequation γ = −∇f γ. Then it is clear that the shifted curve γ • τ = γτ = γ(·+ τ)is also a solution, since

∂

∂t(γ • τ) = γ(·+ τ)

Moreover, it holds that γ(R) = (γ•τ)(R) for all γ ∈M(x, y), that is, the image γ(R)remains unchanged. It is in this sense, of image sets, that we will mean ‘geometrical’behavior of the trajectories.

Proposition 3.29 (Equivalence of the trajectory space definitions. Group action).The additive group R acts smoothly, freely and properly on the manifold M(x, y) by

R×M(x, y)→M(x, y)

(τ, γ)→ γ • τ

provided that x 6= y, so that M(x, y) consists in non-constant trajectories.

Proof. We shall make use of the identificationM(x, y) ≈ W u(x)∩W s(y). To proveit, consider the smooth evaluation map:

E0 : P1,2x,y(R,M)→M

γ → γ(0)

22

Smoothness follow immediately from the representation in suitable local coordinates,

E0,loc : H1,2R (h∗D)→ U(h(0)) ⊂ Th(0)M

ξ →(

exp−1h(0) E0 exph

)(ξ) = ξ(0)

where E0,loc is continously linear. Restricting the evaluation map E0 to the trajectoryspace M(x, y), we obtain an embedding

E0 :M(x, y) →M

because the differential

DE0(γ) : TγM(x, y)→ Tγ(0)M

DE0(γ) · ξ = ξ(0)

is injective at each γ ∈ M(x, y). To see that, let us consider ξ ∈ TγM(x, y) =kerDF (γ), satisfying ξ(0) = 0. As analyzed in the Fredholm section, ξ may betreated as the solution of a linear ordinary differential equation, so that uniquenessof solutions gives us ξ ≡ 0. So the evaluation map leads to a diffeomorphism betweenthe manifolds

E0 :M(x, y)≈→ W u(x) ∩W s(y)

γ → γ(0)

Now, it is obvious that E0 identifies the group action (τ, γ)→ γ•τ with the negativegradient flow,

R× (W u(x) ∩W s(Y ))→ W u(x) ∩W s(y)

(t, p) → Ψt(p)

This gradient flow represents a smooth, free and proper R-action. Hence the equiv-alence γ • τ = (E−1

0 Ψt E0)(γ) finishes the proof.

Now the following definition makes sense:

Definition 3.30 (Unparametrized trajectory spaces). Given (M, g) a riemannianmanifold, f a Morse function on it, and p, q ∈ Crit(f), we define the space ofunparametrized trajectories by the formula:

M(p, q) =M(p, q)/ ∼

where ∼ stands for the relation induced from the action of the group R by translationof the time parameter. In view of the previous results, this quotient is a well-definedmanifold of dimension ind p− ind q − 1.

Now, define the smooth function ϕ = f E0 : M(x, y) → R with dϕ(γ) · ξ =df(γ(0)) · ξ(0) = 〈∇f(γ(0)), ξ(0)〉.

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Definition 3.31. Let f(y) < a < f(x), such that γ(0) is not critical for anyγ ∈ ϕ−1(a). Then a is a regular value of ϕ and

Ma(x, y) = ϕ−1(a)

is a (ind(x)− ind(y)− 1)-dimensional submanifold of M(x, y).

The following proposition gives us an interesting description of the unparametrizedtrajectory spaces:

Proposition 3.32. The map

Ψa : R×Ma(x, y)→M(x, y)

(τ, γ)→ γ • τ

represents an R-equivariant diffeomorphism, with respect with the trivial action onthe left and the above one on the right.

Proof. The R-equivariance Ψa(τ + σ, γ) = Ψa(τ, γ) • σ is obvious from proposition3.29. It is also easy to verify bijectiviy, with 3.29 and the uniqueness of solutions ofordinary differential equations. Thus, it is enough to prove that

DΨ(τ, γ) : R× TγMa(x, y)→ Tγ•τM(x, y)

is an isomorphism. Let (s, ξ) ∈ R × TγMa(x, y). Then the definition of ϕ abovegives us

df (γ(0)) · ξ(0) = 0 (3.9)

and given (s, ξ) ∈ kerDΨa(0, γ), the equation

0 = DΨa(0, γ) · (s, ξ) = γ · s+ ξ

follows. From γ = −∇f γ we obtain ξ(0) = s · ∇f(γ(0)), so that (3.9) impliesξ(0) = 0. So we conclude the injectivity of DΨa(0, γ), and the surjectivity followsby a dimension argument. Now, the identity

Ψa(τ, γ) = φτ (Ψa(0, γ))

finishes the proof, since by proposition 3.29, φτ is a diffeomorphism.

Remark 3.33. This means Ψa is a diffeomorphism

M(x, y) ≡M(x, y) ≈Ma(x, y)

so M(x, y) can be viewed as the transversal intersection of the surface associatedto a regular level of f with W u(x) ∩W s(y).

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3.4.2 Compactification result

We now turn our attention to the main compactness result. For that, it will beconvenient to introduce some a new condition for f .

Definition 3.34 (Palais-Smale condition). A function f fulfills the Palais-Smalecondition, if every sequence (xn)n∈N ⊂ M , such that (|f(xn)|)n∈N is bounded and|∇f(xn)| → 0, has a convergent subsequence.

Remark 3.35. Note that a Morse function on a compact manifold will automaticallysatisfy this condition.

The precise sense of compactness we will be talking about is the following:

Definition 3.36 (Compactness up to broken trajectories). A subset K ⊂ M(x, y)is called compact up to broken trajectorioes of order ν, or up to (ν − 1)-brokentrajectories exactly if, for all (un)n∈N ⊂ K, either:

(1) There exists a convergent subsequence (unk), or

(2) There are critical points

x = y0, . . . , yi = y ∈ Crit(f), 2 ≤ i ≤ ν

and connecting trajectories together with associated reparametrization times

vj ∈Myj ,yj+1, (τn,j)n∈N ⊂ R, j = 0, . . . , i− 1

such that for some subsequence (nk)k∈N, the following convergence holds

unk • τnk,jC∞loc→ vj

Definition 3.37 (Geometrical convergence). The convergence of trajectories above,

wjC∞loc→ w in C∞x,y(R,M) will be called weak convergence in what follows, and the weak

convergence of unparametrized trajectories subject to suitable reparametrizationtimes will be called geometrical convergence.

The main result of this section is the following

Theorem 3.38. [Compactification result]If f satisfies the Palais- Smale condition,

the manifold M(x, y) is compact up to broken trajectories of order µ(x)− µ(y).

This immediately implies the following fact, which will be crucial for constructingMorse homology:

Corollary 3.39. If, moreover, indx− ind y = 1, then M(x, y) is a finite set.

Now we state an easy consequence of the Palais-Smale condition, which we willneed to prove the compactness result.

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Proposition 3.40. If f satisfies the Palais-Smale condition above, there is an ε > 0depending merely on f , x and y, such that the set

Kx,yε = z ∈M | f(y) ≤ f(z) ≤ f(x), ‖∇f(z)‖ ≤ ε

is relatively compact.

Let’s get started. As it is often the case with compactness results, an aplicationof the Arzela and Ascoli theorem is at the heart of the argument:

Lemma 3.41. Every sequence (un)n∈N ⊂ M(x, y) has a weakly convergent subse-quence

unkC∞loc→ v ∈ C∞(R,M)

Proof. The strategy will be to transition from the H1,2 topology of the Hilbertsubmanifold M(x, y) to the C0

loc topology. it will be crucial to have an uniformbound for the H1−2-norm of trajectories with fixed endpoints x, y ∈ Crit(f). Oncewe have C0

loc-convergence, the C∞loc will, in principle, follow from elliptic regularity.Now, the fact that the trajectories un arise from the negative gradient flow, with

fixed endpoints x, y gives us the estimate∫ t

s

|un(τ)|2dτ =

∫ t

s

〈un,−∇f un〉dτ ≤ f(x)− f(y), for all s ≤ t (3.10)

Now let d be the Riemannian metric on M . We get

d(un(t), un(s)) ≤∫ t

s

|un(τ)|dτ ≤√|t− s|

√∫ t

s

|un(τ)|2dτ ≤√|t− s|

√f(x)− f(y)

where the first inequality is by Holder’s inequality, and the second by 3.10. This inturn means equicontinuity of the un in C0(R,M). To be able to apply the theorem ofArzela-Ascoli, we need to obtain pointwise convergence. For now we have a uniformL2 bound on the derivatives, and the Palais-Smale condition. Choose a fixed t0 ∈ R.By proposition 3.40, we can assume without loss of generality that

|(∇f un)(t0)| ≥ ε for all n ∈ N

And due to the asymptotic decrease lim|t|→∞ |∇f(un(t))| = 0 we can find a sequence(tn)n∈N satisfying:

|∇f(un(tn))| = ε and |∇f(un(s))| ≥ ε for all s ∈ [tn, t0] (3.11)

and thus the points un(tn) ∈ Kx,yε . In other words, there is an r0 > 0 such that

d(un(tn), x) < r0 holds for all n ∈ N. Now, using the inequality

d(x, un(t0)) < r0 + ln(t0) with ln(s) =

∫ s

tn

|un(τ)|dτ

we see that the only thing needed for pointwise convergence is a bound for ln(t0).Calculating

dlnds

(τ) = |∇f un(τ)| ≥ ε

26

d(f un)

ds(τ) = −|∇f un(τ)|2

with 3.11, for tn ≤ τ ≤ t0 gives us the estimate

dlnds

(τ) ≤(−1

ε

)d

ds(f un)(τ)

hence we obtain the desired upper bound from

ln(t0) =

∫ t0

tn

dlnds

(τ)dτ ≤ 1

ε

∫ tn

t0

d(f un)

ds(τ)dτ =

f(un(tn))− f(un(t0))

ε

≤ f(x)− f(y)

ε

and, since (un(t))n∈N is a compact set for each fixed t ∈ R, by the Arzela-Ascolitheorem we get a subsequence (unk)k∈N converging on compact intervals. That is,there exists v ∈ C0(R,M) with v(t) = limk→∞ unk(t), such that

unk|[−R,R]C0([−R,R])→ v|[−R,R] for all R ≥ 0

And the fact the trajectories are solutions of the differential equation u = −∇f uimplies the Ck-convergence,

unk |[−R,R]Ck([−R,R])→ v|[−R,R]

iteratively for all k ∈ N.

Lemma 3.42. Let (un)n∈N ⊂M(x, y) be a weakly convergent sequence,

unC∞loc→ v ∈ C∞(R,M)

such that v ∈M(x, y), that is, v is in the same trajectory space. Then the sequence(un) is also H1,2-convergent,

unP1,2x,y→ v ∈M(x, y)

Proof. It consists on showing that the elements un of the sequence converge uni-formly toward y and x when t→ ±∞ respectively. If that is true, proposition 3.12about the asymptotical behavior of the gradient trajectories implies uniform expo-nential asymptotic convergence, which combined with the C∞loc convergence eventu-ally implies H1,2-convergence.

The details are more technical than interesting, and can be found in [Sch].

Proof of theorem 3.38. Now the objective is to pass to the unparametrized trajec-tory spaces, and explore the obstructions to strong H1,2 convergence there. Lemma3.41 gives us weak convergence to a curve v ∈ C∞, and thus to a trajectory of thenegative gradient flow, that is v +∇f v = 0, but it need not be the case that this

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new trajectory is in the same trajectory space. Given any convergent subsequenceobtained from lemma 3.41, local convergence:

unkC∞loc→ v ∈ C∞(R,M) (3.12)

implies the estimate:

f(v(t)) ∈ [f(y), f(x)] for all t ∈ R (3.13)

so the possible trajectory manifolds where such trajetories lie, of the formM(x′, y′)must satisfy:

f(y) ≤ f(y′) ≤ f(x′) ≤ f(x).

Now we deduce that v is actually an element as element of these spaces, from 3.12and the Palais-Smale condition, as follows:

First, by the above restriction for the values f(v(T )), and the equation v =−∇f v: ∫ T

−T|v(s)|2ds = f(v(T ))− f(v(−T )) ≤ f(x)− f(y) for all t ∈ R

so ‖v‖L2 ≤ ∞, which in turn implies (being v continous) that

limt→±∞

∇f(v(t)) = limt→±∞

v(t) = 0 (3.14)

And finally, by combining the Palais-Smale condition of f with 3.13 and 3.14, wededuce the relation v ∈M(x′, y′), where x′, y′ satisfy f(y) ≤ f(y′) ≤ f(x′) ≤ f(x).

Now, we have to look separately at the different cases for x′ and y′. If v ∈M(x, y), the previous lemma gives us strong convegrence and we are finished. Theother cases which lead to splitting up into broken trajectories, that is, without lossof generality,

v ∈M(x, y′) with f(y) < f(y′)

can be trated by suitable reparametrizations. Let us choose a regular level a of f ,with f(y) < a < f(y′) and a reparametrization of unk ,

uk = unk • τk = unk(·+ τk), (τk)k∈N ⊂ R

such that the identity f((unk • τk)(0)) = a holds for all k ∈ N. Using lemma 3.41

again on this sequence (uk), we get another weakly convergent subsequence uklC∞loc→ v,

which satisfiesf(y) ≤ f(v(+∞)) ≤ f(v(−∞)) ≤ f(y′)

Now, iterating these procedures of sorting by the values of f at the ends of the tra-jectories and reparametrizating either provides us, step by step, with subsequencesH1,2-convergent or C∞loc-convergent to constant trajectories, or ‘recovers’ new criticalpoints. But, by the results of the previous section, the indices at the ends of thenon-constant trajectories must be different:

ind(v(j)(−∞))− ind(v(j)(+∞)) > 0

so this iterative process of reparametrization must stop at broken trajectories oforder at most ind(x)− ind(y). The theorem follows.

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3.5 Gluing

In the last section, we saw how the only obstructions to compactness of the trajectoryspaces are broken trajectories, joining the critical points in question, but passingthrough other critical points, in a non-smooth fashion. Now we want to introducea gluing operation to be able to analyze the behavior of critical lines lying near theboundary. Specifically, one defines a map #ρ that maps simply broken trajectories,that is, pairs (u, v) ∈M(x, y)×M(y, z) into the trajectory spaceM(x, z) in a waydepending on the positive parameter ρ, so that when ρ→ +∞ the glued trajectoryapproaches the original one in a certain way.

This is made precise in the following theorem:

Theorem 3.43. Given a compact set of simply broken trajectories K ⊂M(x, y)×M(y, z), there is a lower bound ρK ≥ 0 and a smooth map

# : K × [ρK ,+∞)→M(x, z)

(u, v, ρ)→ u#ρv

satisfying: The map #ρ : K →M(x, z) is an embedding for each gluing parameter

ρ ≥ ρK. Moreover, given a compact set K ⊂ M(x, y)× M(y, z) of unparametrizedtrajectories, # induces an smooth embedding

# : K × [ρK ,∞) → M(x, z)

such that we obtain weak convergence towards the simply broken trajectory

u#ρvC∞loc→ (u, v)

as ρ→ +∞. Conversely, any sequence of unparametrized trajectories converging toa simply broken trajectory lies within the range of such a gluing map #.

Here we won’t go into the details of this construction, since it is quite involved,and we already have almost everything we need for constructing a Morse homologytheory, most importantly, finiteness of the unparametrized trajectory sets of relativeindex one (by compactness), and the orientations to be constructed below.

The gluing operation tells us two things: that every simply broken trajectory thatcould appear as a boundary actually does (note that this wasn’t clear from the com-pactification result), since we can glue them, and that we can embed those brokentrajectories smoothly into the trajectory spaces, hence generating a manifold-with-corners structure. This, combined with coherent orientations would finish theorem3.1, at least the codimension one part, which is the interesting one for our homology.

3.6 Orientation

Right now, we would be ready to build a version of the Morse homology (that is,homology calculations from critical points) with coefficients in Z2, by counting flowlines from a critical point p to critical points q with ind q = ind p− 1. However, tobe able to admit arbitrary coefficient groups (i.e., coefficients in Z, by the universal

29

coefficient theorem) we need to get orientation results for the trajectory spacesinvolved.

Here we will depart from the analytical point of view, using more geometricalmethods, in order to shorten the exposition. The disadvantage is that these methodsare only valid in a finite-dimensional setting, and although that is enough for us, itleaves little room for generalization.

The starting point is the diffeomorphism of proposition 3.29,

M(x, y) ≈ W u(x) ∩W s(y)

The main result that enables us to define orientations is the following:

Proposition 3.44. Let (M, g) and f satisfy the Morse-Smale condition, and letx, y ∈ Crit(f), γ ∈ M(x, y) be given. Under the above identification, and choos-ing an orientation for W u(x), for any point z ∈ γ(R), we have an isomorphism,canonical at the level of orientations:

TW u(x) ∼= Tz(Wu(x) ∩W s(y))⊕ (TzM/TzW

s(y))

∼= TγM(x, y)⊕ Tzγ ⊕ TyW u(y)

Proof. The first isomorphism comes from the Morse-Smale condition.For the second line, the isomorphism

Tz(Wu(x) ∩W s(y)) ∼= TγM(x, y)⊕ Tzγ

is clear from proposition 3.29 and its corollaries. And the isomorphism

TzM/TzWs(y) ∼= TyW

u(y)

is obtained by translating the subspace TyWu(y) ⊂ TqM along γ while keeping it

complementary to TW s(y).

Definition 3.45 (Orientation of M(x, y)). We define an orientation in M(x, y) sothat the isomorphism above is orientation-preserving.

Note that this is dependent on which orientation one starts with, but as long asthe picks are all consistent, (e.g. letting the W u(x) be all oriented as submanifoldsof M , assuming M orientable), it won’t cause problems.

Now, for the case of interest to us, that is, when the critical points have relativeindex one (it will be the only one used later), defining an orientation is equivalent

to choosing a sign for each point in M(x, y). This leads to the following definition:

Definition 3.46. In the above situation, that is, trajectory space orientations in-herited from an orientation for M , we define the integer

#M(x, y)

as the sum of the numbers (±1) assigned to each u ∈ M(x, y) according to theorientation.

Remark 3.47. For our purposes, the orientation chosen does not matter since a globalsign change will eventually give the same results.

This will later form the basis for the Morse differential operator, for the Morsecomplex with groups defined as the free abelian groups with basis the critical pointswith given index.

30

4 Sheaves, cohomology and currents

Here we investigate connections between the topology of a smooth manifold M, andthe algebra of differential forms on it. Specifically, de Rham theorem relating singu-lar and de Rham cohomology, and the isomorphism between de Rham cohomologyand the cohomology of the current sheaf on M.This is all done through basic sheaf theory, which provides a systematic way totrack local data defined on the manifold, and it’s relation with the global data. Thetreatment here is essentially that of [We], chapter II.

4.1 Basics. Presheaves and sheaves

Definition 4.1 (Presheaf). A presheaf F over a topological space X, with valueson a category C consists of:

(a) For each open subset U ⊂ X, an object F(U) in C.

(b) If V ⊂ U , a morphism rUV : F(U) → F(V ). These are called restrictionmorphisms.

Such that the following conditions hold:

(p1) rUU is the identity morphism for all U .

(p2) If W ⊂ V ⊂ U , then rVW rUV = rUW .

Definition 4.2 (Morphism of preseaves). If F and G are presheaves over X, amorphism of presheaves h : F → G, consists of morphisms hU : F(U)→ G(U), suchthat the following diagram commutes, if V ⊂ U ⊂ X:

F(U) −−−→hU

G(U)yrUV yrUVF(V ) −−−→

hVG(V )

Definition 4.3 (Sheaf). A presheaf is called a sheaf if for each covering Ui of anopen set U , that is, U =

⋃i Ui, the following hold:

(s1) If s, t ∈ F(U), and rUUi(s) = rUUi(t) for all i, then s = t.

(s2) If si ∈ F(Ui), and if for Ui ∩ Uj 6= ∅ we have:

rUiUi∩Uj(si) = rUjUi∩Uj(sj)

For all i, then there is an s ∈ F(U) such that, for all i, rUUi(s) = si.

Thus, informally, a presheaf is a way to coherently assign objects in a category toopen subsets of a topological space, and a presheaf is a sheaf when such local objectscan be ‘globalized’ in a certain sense.

31

Example 4.4 (Constant sheaves). Let X be a topological space and G an abeliangroup. If we assign G to each connected open set U ⊂ X, we trivially get a sheafon X. This sheaf will be called the constant sheaf G on X, denoted also by G.

Example 4.5 (Sheaves of differential forms on a differentiable manifold). On adifferentiable manifold M we define the sheaf of differential forms or order k on it,assigning to each open set U ⊂ M the set Λk(U). It is easy to check that thisdefinition fits the conditions to be a presheaf, and a sheaf.

Example 4.6 (A presheaf which is not a sheaf). Define, on the complex plane C,the presheaf F which assigns to each U ⊂ C the algebra of bounded holomorphicfunctions with domain U . Now consider the open disks Uj = z ∈ C | |z| < j,which cover the complex plane. Now let fj ∈ F(Uj) be defined by fj(z) = z. Butthen there can be no f ∈ F(C) such that f |Uj = fj, and thus the second axiomfor sheaves doesn’t hold. In fact, by Liouville’s theorem, F(C) consists of constantfunctions only.

Given a presheaf F over X, a point p ∈ X, and a neighborhood basis at pdenoted by B(p), then it is obvious the images of the sets in said basis, along withthe correspondent restriction maps, form a direct system in C. This motivates thefollowing definition, when C is a category with algebraic structure preserved bydirect limits, such as abelian groups or commutative rings:

Definition 4.7 (Stalk. Germ). Let F be a presheaf over X, p ∈ X. Then wedefine the stalk of F at p as the direct limit of the sets F(U) with x ∈ U , withrespect to the restriction maps rUV . Given s ∈ F(U), we define its germ at x to besx = rUx (s), where rUx : F(U)→ Fx is the natural homomorphism taking an elementto its equivalence class in the direct limit.

Note that in the usual cases, like the sheaf of differentiable real functions on amanifold, this is equivalent to the usual definition of germs.

4.2 Resolutions of sheaves

We now turn to the sequences of sheaves that will allow us to compute cohomology.First we will look at a natural way to make presheaves a natural object, which inturn will allow us to get a sheaf out of an arbitrary presheaf, even if not a sheaf byitself.

Definition 4.8 (Etale space. Section). An etale space over a topological space Xis a topological space Y , together with a continous and surjective map π : Y → Xwhich is also a local homeomorphism. Given an etale space π : Y → X, and anopen set U ⊂ X, a continous map f : U → Y such that π f = 1U is called a sectionover U . The set of such maps is denoted Γ(U, Y ).

Remark 4.9. It’s easy to see that these sets of sections form a sheaf over X, whichin fact is a subsheaf of the sheaf of continous functions from X to Y .

32

Now we will see a way to associate an etale space F to a presheaf F . This willbe done in such a way that if F is also a sheaf, then the sheaf of sections of the etalespace F is another model for F .

The idea is to take the germs of elements of the sheaf to be the points of thenew space, that is, to take F =

⋃x∈XFx. Then the map π : F → X taking each germ

to its basepoint is obviously surjective.It remains to define a suitable topology on this space. For each s ∈ F(U), define

an application s : U → F by s(x) = sx. Note that π s = 1U . We take the setss(U) | s ∈ F(U), U ⊂ X open to be a basis for the topology in F (it is obviousthat this is a base for a topology). Then all the functions s are continous, and infact this is the final topology for this set of functions.

One then defines the sheaf on X which assigns, to each open set U ∈ X, theset of local sections Γ(U, F), which inherits the algebraic structure of the sets F(U)(given that such structure is preserved by direct limits). That this is actually a sheafis seen in the following proposition:

Proposition 4.10 (The associated sheaf). For any presheaf F , the assignmentdefined above is a sheaf. Moreover, if F is a sheaf, then it is isomorphic to F .

Proof. Now define the map τ : F → F by the functions τU : F(U)→ Γ(U, F), withτU(s) = s as defined above. It suffices to check that τU is bijective for each U. Forinjectivity, suppose that there are s1, s1 ∈ F(U) with τU(s1) = τU(s2).

Then, for all x ∈ U , τU(s1)(x) = τU(s2)(x), that is, rUx (s1) = rUx (s2). Butthen, by the definition of direct limit, there exists a neighborhood V of x, such thatrUV (s1) = rUV (s2).

But this is true for each x, so we can produce an open cover⋃ni=1 Ui = X, such

that rUUi(s1) = rUUi(s2) holds for each i. Since F was a sheaf to begin with, we musthave s1 = s2.

Now let’s check surjectivity. Let σ ∈ Γ(U, F). Then for x ∈ U there is aneighborhood V of x, and s ∈ F(V ) such that σ(x) = sx = τV (s)(x). By thedefinition of section of an etale space, if any two sections agree on a point, theyagree on a neghborhood of that point. So let x ∈ X. We have, for some W aneighborhood of x,

σ|W = τV (s)|W = τW (rVW (s))

Since x was arbitrary, we get an open cover⋃ni=1 Ui = U , such that for each i there

exists si ∈ F(Ui), and σ|Ui = τUi(si). Also, we have

τUi(si) = τUj(sj) on Ui ∩ Uj

using the injectivity proved above, this means rUiUi∩Uj(si) = rUjUi∩Uj(sj). Now, since

F is a sheaf and the Ui cover U , we can patch together the si to get s ∈ F(U) suchsthat rUUi = si. Then

τU(s)|Ui = τUi(rUUi

(s)) = τUi(si) = σ|Ui

which in turn gives us τU(s) = σ.

33

In the rest of our discussion, we will assume all sheaves to be of abelian groups,maybe with additional structure (e.g. sheaves of rings).

Definition 4.11 (Exact sequence of sheaves). For A,B, C sheaves of abelian groups,the sequence:

A → B → C

is exact if the induced stalk sequence:

Ax → Bx → Cx

is exact for every x. Equivalently, for each x there is a neghboorhood x ∈ Ux suchthat the sequence A(Ux) → B(Ux) → C(Ux) is exact. That is, exactness needs tohold only locally.

There is an specific type of exact sequence of sheaves which will be particularlyuseful for cohomology:

Definition 4.12 (Sheaf resolution). A resolution of a sheaf F is an exact sequenceof sheaves of the form:

0→ F → F0 → F1 → . . .→ Fn → . . .

The following fairly simple case will be of special interest to us:

Lemma 4.13 (Poincare lemma). Let A ∈ Rn is an open and star shaped, that is,there is a point p ∈ A such that A contains all lines joining p to any other point.Then every closed differential form on A is exact.

Proof. Elementary but somewhat involved application of calculus. Can be found in[Spi].

Remark 4.14. The classical Poincare lemma guarantees that the following is a reso-lution of the constant R sheaf on a differentiable manifold M :

0→ R→ Ω0M

d→ Ω1M

d→ . . .d→ Ωn

M → 0

Where n = dim(M), and ΩkM denotes the sheaf differential forms on M of order k,

that is ΩkM(U) = Λk(U).

In what follows, we will often consider the groups F(X), which in analogy to theassociated etale space, we call the group of global sections of the sheaf F .

4.3 Sheaf cohomology

We will now look at the problem of global exactness of sequences of sheaves, thatis, for an exact sequence of sheaves 0 → A → B → C → 0, when is the inducedsequence

0→ A(X)→ B(X)→ C(X)→ 0 exact?

Part of this is always true, as the following easy proposition shows:

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Proposition 4.15 (Left exactness of global sections). Given an exact sequence of

sheaves 0 → A g→ B h→ C → 0, the induced global section sequence 0 → A(X)gX→

B(X)hX→ C(X)→ 0 is exact at A(X) and B(X).

Proof. To check exactness at A(X), it suffices to note that if there were to be anon-null section s over X in ker gX , then there would be p ∈ X with s(p) 6= 0, andso the germ sp 6= 0 (and sp ∈ ker gp by naturality), contradicting the fact that thefirst sequence is exact.

Now let b ∈ gX(AX). Then, for all x ∈ X, the germ bx = rXx (b) is in kerhx, byassumption. Since x was arbitrary, this means we can find an open cover Ui ofX, such that for each i, rXUi(hX(b)) = hUi(r

XUi

(b)) = 0, and using the first axiom forsheaves, necessarily hX(b) = 0.

The reverse inclusion is proved similarly: one only needs to note that injectivityof the gx makes the choice of local preimages unique, so that they can be gluedtogether with the second sheaf axiom.

Example 4.16. To see that global sections of sheaves need not be right exact,consider the following sequence:

0→ Z i→ O exp→ O∗ → 0

Where O is the sheaf of holomorphic functions on C − 0, and O∗ is the sheafof nonvanishing holomorphic functions on this same set. The map i is the naturalinclusion, and exp : O → O∗ is defined by expU(f)(z) = exp 2πif(z). Note that thegroup structure for sections of O is additive, whereas for sections of O∗ it must bemultiplicative for the above to make sense.

We claim this is an exact sequence of sheaves (that is, at germ level).To check exactness at O∗, let gx be a point at some point x, take a representative

g of gx defined on a sufficiently small, simply-connected neighborhood U of x. Thenwe can choose fx = ( 1

2πilog g)x, for some branch of the logarithm function log, as a

preimage of gx under exp.For the middle term, expx(fx) = 0 (where 0 is the identity element of the group

O∗§) implies, for a representative f of fx in some connected open neighborhood U ofx,

exp 2πif(z) = 1, for z ∈ U

Which implies that f is constant, and in fact, an integer. Thus ker(expx) = Z.

On the other hand, for global sections, the map O(X)exp→ O∗(X) is clearly not

surjective, since there cannot be an holomorphic logarithm function on the wholeC−0, so the global section sequence 0→ Z→ O(X)→ O∗(X)→ 0 is not exact.

This example is already a hint that the obstructions to exactness for globalsections are often of a topological nature.

It is convenient to extend sections of sheaves to closed sets. This is done throughdirect limits, just as for germs. Let F be a sheaf over X, and S ⊂ X a closed subset.The we define:

F(S) = lim−→S⊂U

(U)

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From the point of view of the associated etale space F , this can be identified withthe set of continous sections of F |S = π−1(S). In this line, one can use the notationΓ(S,F). The direct limit construction also gives us restriction maps F(U)→ F(S),just like for stalks.

Definition 4.17 (Soft sheaf). A sheaf F over X is called soft, if for each closedset S ⊂ X, the restriction map F(X) → F(S) is surjective. In other words, if anysection over a closed set can be extended to a section over the whole space.

Theorem 4.18. If A is a soft sheaf, and

0→ A g→ B h→ C → 0

is an exact sequence of sheaves, then the induced sequence:

0→ A(X)gX→ B(X)

hX→ C(X)→ 0

is also exact.

Proof. Let c ∈ C(X). We need to produce b ∈ B(X) such that hX(b) = c. Since thesequence of sheaves is exact, for each x ∈ X there is a neighborhood U of x, andsome b ∈ B(U), such that hU(b) = c|U . Since x was arbitrary, it follows that we canfind an open cover Ui of X and preimages of the restrictions of c to the open setsof the cover. The question is if those sections can be glued together to form a globalsection, b ∈ B(X).

Being X is paracompact, we can find a refinement Si of Ui, locally finiteand such that each Si is a closed set. Consider the set of all pairs (b, S), where S isa union of some of the Si and b is such that hS(b) = c|S. This set can be partiallyordered by the relation (b, S) < (b′, S ′) if S ⊂ S ′ and b′|S = b. The second axiomfor sheaves implies that in this order, every linearly ordered chain has a maximalelement. Thus, by Zorn’s lemma, we can find a maximal pair (b, S), such thathS(B) = c|S.

Suppose that S 6= X. Then there is some Sj ∈ Si such that Sj 6⊂ S. Buth(bj − b) = c − c = 0 in Sj ∩ S. So, by exactness of the global section sequence atthe term B(X) (proved in the above proposition), there is a ∈ A(Sj ∩ S) such thatg(a) − b − bj. Since A is soft, we can extend a to the whole space X. Using thesame notation for the extension, we can define b ∈ B(S ∪ Sj) by setting

b =

b on S

bj + g(a) on Sj

Then, h(b) = c|S∪Sj , hence S was not maximal and we get a contradiction. Thisfinishes the proof.

There is another relevant class of sheaves in which we will be interested:

Definition 4.19 (Fine sheaf). A sheaf F over a paracompact topological space X isfine, if for any locally finite open cover Ui of X, there is a family of sheaf morphismsηi : F → F , such that:

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(a)∑ηi = 1.

(b) ηi(Fx) = 0 for x in some neighborhood of X − Ui.

We call this family ηi a partition of unity of F subordinate to the cover Ui.

Note that this is equivalent to the usual definition for real-valued partitions ofunity on a manifold, for example.

Proposition 4.20. Fine sheaves are soft.

Proof. Let F be a fine sheaf over X, and S ⊂ X a closed set. Let s ∈ F(S). Then,there is a covering Ui of S, each Ui open in X, and sections si ∈ F(Ui) such thats|S∩Ui = si|S∩Ui .

Now let Uo = X − S, s0 = 0, to extend Ui to covering of X. Since X isparacompat, we can assume Ui to be locally finite, so there is a partition of unityηi subordinate to Ui. Then ηi(si) is a section on Ui which is identically zero ona neighborhood of the boundary of Ui, so it may be extended by zero to all of X.Thus we define:

s =∑i

ηi(si)

which is the desired extension of s.

Remark 4.21. Note that on differentiable manifolds, which is the case of interestto us, most of the ‘natural’ sheaves are fine, and therefore soft, by the existence ofthe usual partitions of unity. This will prove to be a very good situation for ourpurposes.

Corollary 4.22. If A and B are soft, and

0→ A→ B → C → 0

is exact, then C is soft.

Proof. Let S be a closed set, and s a section of C over S. Restrict the abovesequence to S. By softness of A we can apply the theorem to get exactness of0→ A(S)→ B(S)→ C(S)→ 0. Hence, we can find a preimage for s and extend itto X by softness of B. Taking it’s image again will give us a suitable extension ofs.

Proposition 4.23 (Global sections of soft sheaves preserve exactness). If

0→ S0 → S1 → . . .Sn → . . .

is an exact sequence of soft sheaves, then the induced global section sequence:

0→ S0(X)→ S1(X)→ . . .Sn(X)→ . . .

is also exact.

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Proof. Let Ki = ker(Si → Si+1) so we can write short exact sequences

0→ Ki → Si → Ki+1 → 0

For i = 1, K1 = imS0, and S0 is soft. Thus, by the theorem, we have the exactsequence

0→ K1(X)→ S1(X)→ K2(X)→ 0

By induction, the corollary above shows that Ki is soft for all i, so we obtain exactsection sequences

0→ Ki(X)→ Si(X)→ Ki+1(X)→ 0

which we can splice together to get the desired exact sequence.

We now turn to the definition of the cohomology groups of a sheaf. Here theconstruction will be done through a canonical soft resolution of the original sheaft.There are other equivalent ways to define these groups, for example as right derivedfunctors of the global section functor, but the one used here is probably the mostelementary.

Definition 4.24 (Sheaf of discontinous sections). Let S be a sheaf over a topologicalspace X, and S π→ X its associated etale space. We define the presheaf C0(S) as:

C0(S)(U) =f : U → S | π f = 1U

That is, the same condition as for sections over U , but without requiring continuity.It is trivial that this is actually a sheaf. Moreover, there is a natural injectionS → C0(S).

Now we can use this construction to get information about S and X. LetF1(S) = C0(S)/S and C1(S) = C0(F1(S)). Now iterate the process by defining:

F i(S) = Ci−1(S)/F i−1(S) and Ci(S) = C0(F i(S)).

By the very definition, we get exact sequences of the form:

0→ S → C0(S)→ F1(S)→ 0

0→ F i(S)→ Ci(S)→ F i+1(S)→ 0

which we can splice together to get the following long exact sequence:

0→ S → C0(S)→ C1(S)→ . . .→ Ck(S)→ . . .

Definition 4.25 (Canonical resolution). Given a sheaf S over a topological spaceX we call the above sequence the canonical resolution of S. We will also use thenotation

0→ S → C∗(S).

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Now we may take global sections in this canonical resolution, to get a sequence:

0→ Γ(X,S)→ Γ(X, C0(S))→ Γ(X, C1(S))→ . . .→ Γ(X, Ck(S))→ . . .

which, as has already been seen, need not be exact. It is precisely the possible lackof exactness of this sequence that gives rise to the sheaf cohomology groups, as seenin the following definition.

Remark 4.26. Given the cumbersomeness of the above notation, we will denoteCk(S) = Γ(X, Ck(S)). The above sequence is then written in the form 0 →Γ(X,S)→ C∗(S).

Definition 4.27 (Cohomology groups of a space with coefficients in a sheaf). Givena sheaf S over a topological space X, the cohomology groups of X with coefficientsin S, denoted Hk(x,S) are defined as:

Hk(X,S) =kerCk → Ck+1

imCk−1 → Ck

That is, the cohomology groups of the sequence of global sections of the canonicalresolution.

Note that softness is preserved through the whole construction, so if S is soft,the global section sequence would be exact everywhere, and the cohomology groupswould vanish.

Theorem 4.28 (Basic properties and functoriality of the cohomology groups). LetX be a paracompact Hausdorff space. Then the following hold:

(a) For any sheaf S, H0(X,S) = S(X). If S is soft, Hq(X,S) = 0 for q > 0.

(b) (functoriality) for any sheaf morphism h : A → B, there is for each q a grouphomomorphism hq : Hq(X,A)→ Hq(X,B) with the following properties:

(i) h0 = hX : A(X)→ B(X).

(ii) hq is the identity map for all q > 0 if h is the identity map

(iii) gq hq = (g h)q for all q ≥ 0, for g and h sheaf morphisms.

(c) (exact sequences) For any exact sequence of sheaves

0→ A→ B → C → 0

for each q ≥ 0 there is a group homomorphism

δq : Hq(X, C)→ Hq+1(X,A)

such that

(i) The following induced sequence is exact:

0→ H0(X,A)→ H0(X,B)→ H0(X, C) δ0

→ H1(X,A)→ . . .

. . .→ Hq(X,A)→ Hq(X,B)→ Hq(X, C) δq→ . . .

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(ii) A commutative diagram

0 −−−→ A −−−→ B −−−→ C −−−→ 0y y y y y0 −−−→ A′ −−−→ B′ −−−→ C ′ −−−→ 0

Gives rise to a commutative diagram:

0 −−−→ H0(X,A) −−−→ H0(X,B) −−−→ H0(X, C) −−−→ H1(X,A) −−−→ . . .y y y y y0 −−−→ H0(X,A′) −−−→ H0(X,B′) −−−→ H0(X, C ′) −−−→ H1(X,A′) −−−→ . . .

Proof. For part (a)(i), note that the resolution

0→ Γ(X,S)→ C0(X,S)→ C1(X,S)→ . . .

is exact at C0(X,S), so Γ(X,S) = ker(C0(X,S)→ C1(X,S)) = H0(X,S).Part (a)(ii) follows easily from proposition 4.23.For (b) and (c) we will first show that a map h : A → B induces a natural cochainmap h∗ : C(A) → C(B), where C(A) = C(X,A) and C(B) = C(X,B). First wedefine a map

h0 : C0(A)→ C0(B)

by letting h0(sx) = (hs)x, for s a discontinous section of A. This induces a quotientmap:

h0 : C0(A)/A → C0(B)/Bgiven that C0(A)/A = F1(A) and C0(B)/B = F1(B), repeating the above construc-tion gives us a map:

h1 : C0(F1(A))→ C0(F1(B))

wich again induces a map of quotients h1, and keeping on in this way we get, foreach q ≥ 0, a map

hq : Cq(A)→ Cq(B)

so the induced section maps give us the desired map h∗. Given the way it wasconstructed, this map is clearly functorial.Now if

0→ A→ B → C → 0

is exact, then0→ C∗(A)→ C∗(B)→ C∗(C)→ 0

is an exact sequence of complexes of sheaves. But these sheaves are all soft, so theinduced global section maps make

0→ C∗(A)→ C∗(B)→ C∗(C)→ 0

an exact sequence of cochain complexes of abelian groups. We can then derive(via the zig-zag lemma of basic homological algebra) a long exact sequence for thecorresponding cohomology groups:

. . .→ Hq(C∗(A))→ Hq(C∗(B))→ Hq(C∗(C)) δq→ Hq+1(C∗(A))→ . . .

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where δq is constructed in the usual way. Now one just needs to note that theconstruction is natural to get parts (b) and (c).

Definition 4.29 (Acyclic resolution). A resolution of a sheaf S over a space X,of the form 0 → S → A∗ is called acyclic, if for q > 0 and p ≥ 0, all the groupsHq(X,Ap) vanish.

The following theorem is the core result for our purposes, and a very powerfultool in calculation of (co)homology.

Theorem 4.30 (Abstract de Rham theorem). Let S be a sheaf over a topologi-cal space X, and let 0 → S → A∗ be a resolution of S. Then there are naturalhomomorphisms

γp : Hp(Γ(X,A∗))→ Hp(X,S)

where Hp(Γ(X,A∗)) are the cohomology groups of the cochain complex Γ(X,A∗),where the maps are the global section maps derived from the resolution. Also, if theresolution is acyclic, the maps γp are isomorphisms.

Proof. Let Kp = ker(Ap → Ap+1) = im(Ap−1 → Ap) (note that these are sheaves,not global sections, so equality holds). This way K0 = S. We have short exactsequences:

0→ Kp−1 → Ap → Kp → 0

which induce, by the properties of the cohomology groups, exact sequences of theform:

0→ Γ(X,Kp−1)→ Γ(X,Ap)→ Γ(X,Kp)→ H1(X,Kp−1)→ H1(X,Ap)→ . . .

Note that, by definition of the Kp, we have Γ(X,Kp) ∼= ker(Γ(X,Ap)→ Γ(X,Ap+1),and in turn

Hp(Γ(X,A∗)) ∼= Γ(X,Kp)/ im(Γ(X,Ap−1)→ Γ(X,Kp))

So we can define, through the exact sequence above, a map γp1 : Hp(Γ(X,A∗)) →H1(X,Kp−1) which is, in fact, injective. Moreover, if the resolution is acyclic,H1(X,Ap−1) = 0 and, again by the long exact sequence, γp1 is an isomorphism.

Corollary 4.31. If we have an homomorphism of resolutions of sheaves, of theform:

0 −−−→ S −−−→ A∗y yf yg0 −−−→ J −−−→ B∗

then there is an induced homomorphism

Hp(Γ(X,A∗)) gp→ Hp(Γ(X,B∗))

which if f is an isomorphism of sheaves and the resolution are acyclic, is an iso-morphism.

This means that we can calculate sheaf cohomology of a space X with coeffi-cientes in a sheaf S using any acyclic resolution of S, in particular any soft resolution.

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4.4 de Rham theorem

We now use the last result of the previous section, abstract de Rham theorem, torelate different notions of cohomology of a manifold, through soft resolutions andsheaf cohomology. We will start by formulating singular cohomology in terms ofsheaves.

Definition 4.32 (Sheaves of singular cochains). Let M be a differentiable manifold,and for each open set U ⊂M , consider the group of singular cochains defined in Uwith real coefficients Sp(U,R) = HomZ(Sp(U,Z),R), where (Sp(U,Z) is the usualgroup of singular chains of degree p with integer coefficients. Then the assignmentU → Sp(U,R) is easily seen to be a presheaf. We denote the sheaf generated by thispresheaf by Sp(M,R).

Now note that the sequence of sheaves

0→ Gδ→ S0(M,R)

δ→ S1(M,R)→ . . .

is exact, since singular cohomology for an euclidean ball is zero for p > 0, similarlyto the Poincare lemma mentioned above.

Remark 4.33. It’s not hard to see that in our setting, that is, smooth manifolds, wecan use C∞ cochains and get the same results above. This is what we will use fromnow on, Denoting these sheaves Sp∞(M,R)

Lemma 4.34 (Sheaves of modules over soft sheaves of rings are soft). If M is asheaf of modules over a soft sheaf of rings R, then M is itself soft.

Proof. Let s ∈ Γ(K,M), where K ⊂ X is a closed subset. Then, by the directlimit definition of such sections, s extends to an open set U ⊃ K. Now defineρ ∈ Γ(K ∪ (X − U),R) be defined by:

ρ =

1 on K

0 on X − U

Then, since R is soft, we can extend ρ to a section over the whole of X, and keepingthe same notation for the extension, ρ · s is the desired extension of s.

Which inmediately gives us what we were looking for, taking into account thatthe constant sheaf R is obviously soft:

Corollary 4.35 (Softness of the singular cochain sheaf). Let M be a differentablemanifold. Then the sheaves of C∞ singular cochains, Sq∞(M,R), are soft.

Remark 4.36. One can define an homomorphism I : Ω∗M → S∗∞(M,R), and, inturn an homomorphism of the corresponding resolutions of the sheaf R, By integra-tion of differential forms over smooth chains. That is, by defining IU : Ω∗M(U) →S∗∞(M,R)(U) by the equation:

IU(φ)(c) =

∫c

φ

where c is a C∞ singular chain, so IU(φ) is a C∞ singular cochain on U. Moreover,Stokes’ theorem implies that I commutes with the corresponding differentials.

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We have already seen that the differential forms on a differentiable manifold Mform a soft resolution of the constant sheaf R. Therefore, we have obtained thefollowing well-known theorem.

Theorem 4.37 (de Rham theorem). Let M be a differentiable manifold. Then thenatural maps I : Hp(Ω∗M(M)) → Hp(S∗∞(M,R)) induced by integration of differen-tial forms over C∞ singular cochains with real coefficients are all isomorphisms.

Proof. That the resolutions induce isomorphisms of their cohomology groups withthe cohomology groups of M with coefficients in R follows from the ‘abstract deRham’ theorem 4.30 above, and the fact that the composite isomorphism is inducedby the integration maps follows from corollary 4.31.

4.5 Currents and homology

Now we turn our attention to a certain kind of duals of the differential form groups.By analogy to singular homology and cohomology, and looking at the de Rhamtheorem above, one would expect this new groups to be useful in calculating singularhomology of the underlying manifold. In fact, even more is true, as we shall seebelow.

Definition 4.38 (Currents). Let M be a smooth manifold and let Λkc (M) be the

space of differentiable k-forms with compact support on it. A k-current is a linearfunctional on Λk

c (M), which is continous in the sense of distributions. The space ofk-currents on M will be denoted Dk(M), endowed with the weak-* topology.

Example 4.39. Let M be a differentiable manifold, m = dim(M), β ∈ Λk(M).Then we can associate to β a m− k-current defined by

Tβ(α) =

∫M

β ∧ α

and the map β → Tβ is injective.

Since it will not be needed for our purposes, we won’t get into analytical detailsabout currents. As expected, one can form a presheaf of currents by taking currentsdefined on open sets U ⊂ M (note that the compact support requirement stillapplies), that is, the assignment U → Dk(U). It is not hard to check that this isalso a sheaf, by standard arguments. This sheaf will be denoted DMk .Now, analogously to the differential form and singular cases, one considers the maps:

DMn∂n→ DMn−1

∂n−1→ . . .→ DM0 → 0

(where n = dim(M)) dual to the exterior derivative of differential forms, that is,defined by ∂kT (ω) = T (dω). Thus, one gets immediately ∂k ∂k−1 = 0, that is, theabove is a differential complex of sheaves.

Definition 4.40 (Pushforward of currents). Let M1,M2 be differentiable manifoldsof dimensions m1 and m2 respectively, and F : M1 → M2 a smooth map. Thenthere is a pullback morphism of forms, F ∗ : Λk

c (M2) → Λk(M1) which is continous

43

in the C∞ topology, but for α ∈ Λkc (M2), SuppF ∗α ⊂ F−1(Suppα) need not be

compact.If T ∈ Dk(M1) is such that for every compact subset K of M2, SuppT ∩F−1(K)

is compact, then α→ T (F ∗α) is well defined and continuous on Λkc (M2), so there is

a unique current F∗T defined by the equation

F∗T (α) = T (F ∗α)

which we call the direct image or pushforward of T by F .

We will need the following easy properties:

Proposition 4.41 (Properties of pushforwards of currents). Let M1,M2,M3 be dif-

ferentiable manifolds, and M1F→ M2

G→ M3 smooth maps. Then, for suitable cur-rents in the sense of the previous definition, the following hold:

(a) ∂(F∗T ) = F∗∂T

(b) (G F )∗T = G∗(F∗T )

Proposition 4.42 (Poincare lemma for currents). Let U ⊂ Rm be star-shaped andopen, and T ∈ Dq(U) be such that ∂qT = 0. Then there is a (q + 1)-currentS ∈ Dq+1(U) such that ∂q+1S = T .

Proof. The trick is to reduce it to the standard Poincare lemma for differential forms,by standard distribution techniques (in essence, if a current is regular enough it canbe represented by a form, just like for functions and distributions). Details can befound in [De], albeit with a somewhat different notation.

Remark 4.43. The above proposition shows that one can form a resolution of theconstant sheaf R of the form:

0→ R→ DMn∂n→ DMn−1

∂n−1→ . . .→ DM0 → 0

Proposition 4.44 (Resolution with currents is acyclic). Let M be a compact smoothmanifold. Then the resolution above is acyclic.

Proof. It suffices to prove that each DMk is soft. But partitions of unity on M ensurethat these sheaves are fine, hence soft.

We have finally arrived to the main theorem we wanted to prove in this section:

Theorem 4.45 (Singular homology from currents). Let M be a compact smoothorientable manifold. Then the singular homology groups of M are isomorphic to thehomology groups of the complex

DMn (M)∂n→ DMn−1(M)

∂n−1→ . . .→ DM0 (M)→ 0

of global sections of the sheaves of currents on M . That is, Hk(M) ∼= Hk(DM∗ (M)).

44

Proof. One just needs to note that

0→ R→ DMn∂n→ DMn−1

∂n−1→ . . .→ DM0 → 0

is an acyclic resolution of the constant sheaf R, so that the complex in the statementof the theorem calculates singular cohomology, by the ‘abstract de Rham’ theorem4.30, and then apply Poincare duality.

A very geometrical explanation and proof of Poincare duality can be found inthe last chapter of [Mu]. A quick introduction to currents can be found on [De], andfurther material in [dR], for example.

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5 The Morse homology theorem

In this section we put together the material from the previous ones, and prove thetheorem we have been aiming for, that is, that the singular homology groups of asmooth manifold are isomorphic to the homology groups of a certain chain complex,constructed with the free abelian groups with basis the critical points of each index.The analytic constructions in section 3 already pointed out how the boundary mapsshould be constructed. This construction is formalized below.

Remark 5.1. Unless otherwise noted, in this section all differentiable manifolds willbe regarded as compact and orientable.

5.1 The Morse chain complex

Definition 5.2. Let (N, g) a riemannian manifold, f : N → R a Morse function,and Criti(f) the set of index i critical points of f . Then we define the groupsCMorsei (f, g) =

⊕p∈Criti(f) Z, and the corresponding differential ∂Morse given by the

following equation:

∂Morse(p) =∑

q∈Criti−1(f)

#M(p, q) · q

where #M(p, q) denotes the (signed according to orientation) number of components

of M(p, q), or alternatively, the number of points in M(p, q), which must be finitesince ind p− ind q = 1.

Now, all the work in section 3 already starts to be fruitful.

Proposition 5.3. The groups and maps defined above form a chain complex. Thatis, (∂Morse)2 = 0.

Proof. This is an easy consequence of theorem 3.1. One just needs to compute:

〈(∂Morse)2p, q〉 =∑

r∈Criti−1(f)

〈∂Morsep, r〉〈∂Morser, q〉 =

= #⋃

r∈Criti−1(f)

M(p, r)×M(r, q)

= #∂M(p, q)

= 0.

Where the second equality is by definition, the third by theorem 3.1, and the fourthbecause the boundary of a compact manifold of dimension 1 will always have zeropoints, counted with sign.

In analogy with the notation above, we will denote the homology groups of thiscomplex HM

k (f, g).

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5.2 The chain homotopy. Morse homology theorem

The goal of this part (and the essence of the whole text) is to prove the followingtheorem:

Theorem 5.4 (The Morse homology theorem). Let (N, g) a compact riemannianmanifold and f : M → R a Morse function, such that (f, g) is a Morse-smale pair.Then there is a canonical isomorphism

HMk (f, g) ∼= Hk(N,Z)

where Hk(N,Z) denotes the k-th singular homology group of N with coefficients inZ.

The proof will consist in producing a chain homotopy between the two chaincomplexes, which will in turn induce homology isomorphisms. The forward mapwill consist in taking a critical point to it’s descending (unstable) manifold, and thebackwards one in flowing a simplex with the gradient flow and taking it to the sumof the critical points it ‘reaches’ at ±∞. To make this all rigorous we will need todefine some concepts and prove a number of things about them first.

Definition 5.5. An i-simplex σ : ∆i → N is said to be generic, if σ is smooth,and each face of σ is transverse to the ascending manifolds of all the critical pointsof f . We define the groups Ci(N) to be the Z-module with basis consisting on thei-currents on N generated by generic i-simplices.

Remark 5.6. What we mean above by equating simplices and currents, is the inclu-sion j : Sk(N,R)→ Dk(N) defined by

j(σ)(α) =

∫σ

α

For σ a singular simplex ∈ Sk(N,Z) and then extending to chains, where Sk(N,R)is the usual singular chain group with R coefficients, and α ∈ Λk(N). Often we willidentify σ and j(σ).

Remark 5.7. Analogously to the above, one can consider, for an oriented submanifoldL ⊂ N of dimension k, the k-current [L] defined by

[L](α) =

∫L

α

Proposition 5.8. (a) In the groups Ck(N), the standard simplex boundary definesthe same maps as the current boundaries ∂k : Dk → Dk−1.

(b) Considering the chain complex (C∗(N), ∂∗), there is a canonical isomorphismof its homology groups with the singular homology groups of N

Hk(C∗(N)) ∼= Hk(N,Z)

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Proof. Part (a) is a direct consequence of Stokes’ theorem.For part (b), one could consider the groups Ck(U) = hom(Ck(U),Z) (where the

transversality condition is kept by restricting f to U) and follow the same stepsused to prove the de Rham theorem from the ‘abstract de Rham’ theorem. Thatwas done with the sheaves S∗∞(M,R).

In this case, one would start defining presheaves U → Ck(U), then taking thesheaves generated from those presheaves (the transversality condition is a local oneso it does not interfere), and considering the corresponding resolution of the constantsheaf Z. It will indeed be a resolution, given that exactness won’t be affected bythe small perturbations that might be needed to fulfill the transversality conditions(we won’t give further details here). Softness of these sheaves is obvious by theirdefinition. One would then have Hk(C∗(N)) ∼= Hk(N,Z). This in turn impliesHk(C∗(N)) ∼= Hk(N,Z).

For our purposes, we will need the following generalized version of theorem 3.1.We don’t give the details, but it can be proved analogously to the latter.

Theorem 5.9 (Compactification of descending manifolds). For a closed smoothmanifold N , (f, g) a Morse-Smale pair on N , and a critical point p ∈ M , thedescending manifold W u(p) has a natural compactification to a smooth manifoldwith corners W u(p), whose codimension k stratum is

W u(p)k =⋃

q1,...,qk∈Crit(f)

M(p, q1)×M(q1, q2)× . . .×W u(qk)

with r1, . . . , rk, q all different. For the case k = 1, as oriented manifolds, we have:

∂W u(p) =⋃

p 6=qCrit(f)

(−1)ind p+ind q+1M(p, q)×W u(q)

and the maps W u(p)k → N given by projecting to W u(qk) patch together to a smoothmap:

e : W u(p)→ N

extending the inclusion W u(p)→ N

Definition 5.10 (D map). We define the map D : CMorsek (f, g)→ Ck(N), by letting,

for p ∈ Critk(f), D(p) = e∗[W u(p)], where e is the embedding map above, so e∗ isthe corresponding push-forward of currents.

Remark 5.11. Note that when using currents, signs are implicitly included, sinceintegration is done on oriented submanifolds.

Proposition 5.12. D is a chain map, that is, ∂ D = D ∂Morse

Proof. Let p ∈ Criti(f). By the last compactification theorem, we have:

∂W u(p) =⋃

q∈Crit f

q 6=p

(−1)ind(p)+ind(q)+1M(p, q)×W u(q)

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Therefore

∂D(p) =∑

q∈Crit f

q 6=p

(−1)ind(p)+ind(q)+1e∗[M(p, q)×W u(q)] ∈ Ci−1(N)

But if ind(q) > i− 1, being f Morse-Smale,M(p, q) is empty, and if ind(q) < i− 1,then the contribution to the right hand side is zero in Ci−1(N), because e wouldmap M(p, q) × D(q) top the support of D(q), which is a current of dimension lessthan i− 1. We then have:

∂D(p) =∑

q∈Criti−1 f

#M(p, q) · e∗[W u(q)]

For the inverse map, we must consider flow lines originating from a given simplex,so one defines

M(σ, q) =

x ∈ N | lim

t→+∞γx(t) = q and for some t0, γx(t0) ∈ σ

for a simplex σ and a critical point q. Note that the support of the simplex hasbeen denoted in the same way, to simplify notation. Notice that by the Morse-Smalecondition there is an isomorphism, for any such x

Tγx(t0)σ ∼= TγM(σ, q)⊕ TqW u(q)

and so we give an orientation to TγM(σ, q) in such a way that this isomorphism isorientation-preserving. We require yet another compactification theorem for thesesets, which again can be proved in the same way as the previous ones.

Theorem 5.13 (Compactification of M(σ, q)). With the notation above, M(p, q)has a natural compactification to a smooth manifold with corners M(σ, q), withcodimension k stratum

M(σ, q)k =k⋃j=0

⋃p1,...,pk∈Crit(f)

p1,...,pk,q distinct

M(σk−j, p1)×M(p1, p2)× . . .×M(pj−1, pj)×M(pj, q)

where σj denotes the codimension j stratum of σ. When k = 1, as oriented manifoldswe have:

∂M(σ, q) =M(∂σ, q) ∪⋃

p∈Crit(f)

p 6=q

(−1)i+ind(q)M(σ, p)×M(p, q)

Taking into accout the above theorem, and that dimM(σ, q) = i − ind(p), onecan define a map A : C∗(N)→ CMorse

∗ (f, g) through the formula

A(σ) =∑

p∈Criti(f)

#M(σ, p) · p

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Proposition 5.14 (A is a chain map). A ∂ = ∂Morse A

Proof. Like before, this follows from the above compactness result.Let q ∈ Criti−1(f). Then:

#∂M(σ, q) = #M(∂σ, q)−#⋃

q 6=p∈Crit(f)

M(σ, p)×M(p, q)

= #M(∂σ, q)−#⋃

q∈Criti f

M(σ, p)×M(p, q)

= 〈A(∂σ), q〉 − 〈∂MorseA(σ, q)〉

Lemma 5.15. A D = id : CMorsei → CMorse

i

Proof. If p ∈ Criti(f), then M(D(p), p) contains one point, the constant flow line,oriented positively. On the other handM(D(p), q) is empty if q is any other index icritical point, seeing the definition of D(p), and by the Morse-Smale condition.

We will need a last compactification theorem:

Theorem 5.16 (Compactification of forward orbits). For σ a generic i-simplex, wedefine its forward orbit to be the set

F(σ) = [0,+∞)× σ

together with the map e : F(σ)→ N defined by

e(s, x) = γs(σ(x))

This forward orbit has a natural compactification to a smooth manifold with cornersF(σ), whose codimension k stratum, for k > 2, is

F(σ)k = F(σk)∪k⋃j=0

⋃r1,...,rk∈Crit(f)

r1,...,rk distinct

M(σk−j, r1)×M(r1, r2)×. . .×M(rj−1, rj)×W u(rj)

For k = 1, as oriented manifold one has

∂F(σ) = −σ ∪ −F(∂σ) ∪⋃

r∈Crit(f)

M(σ, r)×W u(r)

And the map e extends to this compatification as a smooth map which projects toW u(rj) ⊂ N .

And this theorem enables us to get what we were looking for.

Proof of the Morse homology theorem. Define F : Ci(N)→ Ci+1(N) by

F (σ) = e∗[F(σ)]

Then the above compactification result implies that F is a chain homotopy betweenthe identity and D A,

∂F + F∂ = D A− id

and by basic homological algebra, we are done.

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5.3 The Morse inequalities

We now present some applications of the Morse homology theorem, most notablythe so-called Morse inequalities. It is worth noting that most or all of these resultscould be proved more directly without needing the full power of the Morse homologytheorem, but here we will deduce them from it.

First, we will settle some notation. Let N be a compact smooth manifold andlet bk denote the k-th Betti number of N , that is, the rank of the finitely gen-erated Z-module Hk(N,Z). Now, for a Morse function f : N → R, let νk(f) =Card(Critk(f)). Note that for chain complexes and boundaries we will use the samesymbols as in the previous part, but with different meanings. Hopefully the meaningwill be clear.

An obvious consequence of the Morse homology theorem is the inequality νk(f) ≥bk, since the Morse complex with the critical points as generators has the samehomology. These are called the weak Morse inequalities. Somewhat more interestingare the following stronger inequalities:

Theorem 5.17 (The Morse inequalities). With the above notation,

(a)∑n

k=0(−1)k+nνk(f) ≥∑n

k=0(−1)k+nbk

(b)∑n

k=0(−1)kνk(f) =∑n

k=0(−1)kbk

where n = dim(N).

To prove this, we will need the following:

Theorem 5.18 (Euler-Poincare theorem). Let (C∗, δ∗) be a finitely generated chaincomplex, for which Ck = 0 if k > n, for a certain n > 0. Let ck = Rank(Ck),bk = Rank(Hk(C∗)), for k = 0, . . . , n. Then the following equation holds:

m∑k=0

(−1)kck =m∑k=0

(−1)kbk

Proof. The exactness of

0→ ker ∂k → Ck∂k→ im ∂k → 0

shows that ck = Rank(ker ∂k) + Rank(im ∂k) for each k. Similarly,

0→ im ∂k+1 → ker ∂k → Hk(C∗)→ 0

being exact gives us Rank(ker ∂k) = Rank(im ∂k+1) + bk, hence

Rank(ker ∂k) = ck − Rank(im ∂k) = Rank(im ∂k+1) + bk

which in turn implies:m∑k=0

(−1)k(ck − Rank(im ∂k)) =m∑k=0

(−1)k(Rank(im ∂k+1) + bk)

which in turn impliesm∑k=0

(−1)kck =m∑k=0

(−1)kbk

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Proof (of the Morse inequalities). Since by the Morse homology theorem, the Morsecomplex has Hk(N,Z) as homology, we can apply the above theorem to immediatelyget the second part of the theorem. For the first part, let m < n and consider thetruncated Morse chain complexes (C

(m)∗ , ∂k) given by:

C(m)k =

CMk (f, g) if k ≤ m

0 if k > m

where a Morse function f and suitable Riemannian metric g are assumed. Then,applying the Euler-Poincare theorem we have

(−1)mm∑k=0

(−1)k Rank(C(m)k ) = (−1)m

m∑k=0

(−1)k Rank(Hk(C(m)∗ ))

Now, since νk(f) = Rank(C(m)k ) for k ≤ m, bk = Rank(Hk(C

(m)∗ )) for k ≤ m − 1,

and Hm(CM∗ (f, g)) is a quotient of Hm(C

(m)∗ ), we end up with

νm − νm−1 + . . .+ (−1)mν0 ≥ bm − bm−1 + . . .+ (−1)mb0

as desired.

An easy application of the Morse inequalities is the following proposition:

Proposition 5.19. The Euler characteristic χ(N) of a compact smooth manifoldN of odd dimension n is zero.

Proof. Let f : N → R be a Morse function (which we know always exist). Trivially,νk(f) = νn−k(−f). Then we have the following:

χ(N) =n∑k=0

(−1)kνk(f) =n∑k=0

(−1)kνn−k(−f) =

= (−1)nn∑k=0

(−1)n−kνn−k(−f) = (−1)nn∑k=0

νk(−f) = (−1)nχ(N)

And since n is odd, χ(N) = 0.

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6 Generalizations

Here we give some ideas (without any sort of proofs) about some generalizations ofthe theory presented.

6.1 Morse-Bott theory

The fact that a small perturbation will make any smooth function Morse seen insection 2 sounds good in principle. But in an specific situation, there is no obviousway of choosing a Morse function for a given manifold, generic as they may be.Especially since choosing a function with obvious symmetries will often work againstus, like the height function of a lying torus, which has critical circles, instead ofisolated points.

One way to go around this is to extend the theory to Morse-Bott functions. AMorse-Bott function is a smooth function M → R such that Crit(f) is a union ofsubmanifolds, and everywhere on those submanifolds the kernel of the Hessian isexactly the tangent space to such critical submanifolds.

The theory then follows in much the same lines, except that one has to considermore complex moduli spaces (informally, one has to ‘flow down’ bigger subsets).Using appropiate metrics and defining a similar chain complex with the criticalsubmanifolds, and taking extra care of the indices involved, the homology of M isrecovered, just like in the Morse function case.

This theory is presented, for example, in [La], with a method based on currentsand spectral sequences, rather than analysis. Some of the development can also befound in [Hu], using the familiar compactification approach.

6.2 Novikov homology

Another way for generalization is to note that after the first definitions, everythingis based on the closed 1-form df , rather than on the actual function f . This suggestsreplacing df by a different closed 1-form α, and trying to repeat the process. This,actually, turns out to be very fruitful.

The process of constructing such an homology is esentially the same, one definesMorse forms to be the ones with nondegenerate ‘critical points’ (that is, zeroes) andan analog of the gradient for such a form raising the index of α through a Riemannianmetric g. The compacntess theorems can be then proved with very similar argumentsto the Morse function case, the main difference being that one has to ‘mod out’ the(de Rham) cohomology class of α throughout. Once one obtains finite counts of therelevant flow lines, the construction of the chain complex is essentially the same.

Now, the interesting part is that, unlike in the original case of a real Morse func-tion, the forms used might have non-zero homology classes. This can be most easilyseen if one starts with functions f : M → S1, in what is called circle-valued Morsehomology. To understand what happens in these cases, it is useful to remember theGalois correspondence of covers of M and subgroups of its fundamental group, andthe relation between the fundamental group and H1(M,Z).

In the end, what one gets is the homology of a covering space M →M in whichthe lift of α is exact, with coefficients in a certain (Novikov) ring defined based on

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the algebraic properties of the class [α] ∈ H1(M,Z). This is useful in a number ofsituations, incluiding the study of periodic orbits of the gradient flow, which in thecase of circle-valued Morse functions, are not excluded.

A good reference for this theory and many of its consequences is [Paj].

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[Br] Bredon, Glen E. Topology and geometry, Graduate Texts in Mathematics 139,Springer-Verlag 1993.

[Br2] Bredon, Glen E. Sheaf theory, Graduate Texts in Mathematics 170, 2ed.,Springer-Verlag 1997.

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[dR] de Rham, Georges. Differentiable manifolds, Springer-Verlag 1984.

[Flo] Floer, A. The unrgularized gradient flow of the symplectic action, Comm. PureAppl. Math. 41 (1988), 393-407.

[GuiPo] Guillemin V., Pollack A. Differential Topology, Prentice Hall, 1974.

[Hu] Hutchings, Michael. Lecture notes on Morse homology (with an eye towardsFloer theory and pseudoholomorphic curves),http://math.berkeley.edu/∼hutching/teach/276/mfp.ps

[La] Latschev, Janko. Gradient flows of Morse-Bott functions, Math. Ann. 318(2000), no. 4, 731-759.

[Lan] Lang, S. Differential and Riemannian Manifolds, Graduate Texts in Mathe-matics, 160. Springer-Verlag 1995.

[Mi] Milnor, John. Morse Theory, Princeton University Press 1963.

[Mu] Munkres, James R. Elements of algebraic topology, Perseus 1984.

[Paj] Pajitnov, Andrei V. Circle-valued Morse theory, de Gruyter 2006.

[Sch] Schwarz, Matthias. Morse Homology, Birkhauser 1993.

[Sm] Smale, Stephen. An infinite dimensional version of Sard’s theorem, AmericanJournal of Mathematics, Vol. 87, No. 4 (Oct., 1965), pp. 861-866

[Spi] Spivak, M. Calculus on manifolds, Addison-Wesley 1965.

[We] Wells, Raymond O. Jr. Differential Analysis on Complex Manifolds, GraduateTexts in Mathematics. 3ed., Springer-Verlag 2008.

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