higher index theory: a survey. · plan of the talk: 1 dirac operators 2 atiyah-singer index theory...

Higher index theory: a survey.

Paolo Piazza (Sapienza Universita di Roma)

Incontri di geometria noncommutativaNapoli, Settembre 2012.

Paolo Piazza (Sapienza Universita di Roma) () Higher index theory: a survey.Incontri di geometria noncommutativa Napoli, Settembre 2012. 1

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Plan of the talk:

1 Dirac operators

2 Atiyah-Singer index theory

3 Eta invariants and rho-invariants

4 Atiyah-Patodi-Singer index theory

5 Primary versus secondary invariants

6 A hierarchy of geometric structures

7 Higher index theory

8 Applications


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Dirac-type operators.

We consider a riemannian manifold (M, g) without boundary and aDirac type operator D : C∞(M,E )→ C∞(M,E )

Example: M is spin and E is the spinor bundle.

Recall that a Dirac-type operator D is defined by a hermitian complexbundle E endowed with a connection ∇E and Clifford action c ,C∞(M,T ∗M ⊗ E )

c→ C∞(M,E )

by definition an operator of Dirac type is obtained taking the

composition C∞(M,E )∇E

→ C∞(M,T ∗M ⊗ E )c→ C∞(M,E ).

thus D := c ◦ ∇E .

we assume the Clifford action to be unitary and the connection on Eto be metric-compatible ⇒ D = D∗ (examples in a moment)


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Basic properties of Dirac operators.

D is an elliptic differential operator

hence if M is compact without boundary, then D is Fredholm

this means that the dimension of the kernel and the cokernel is finite

the index of a Fredholm operator P is by definition ind P ∈ Z =dim ker P − dim cokerP = dim ker P − dim ker P∗

if dim M = 2k then E is graded, E = E+ ⊕ E− and D is odd:

D =

(0 D−

D+ 0

)D− = (D+)∗

if dim M = 2k, ind(D) = 0 (since D = D∗), but ind D+ 6= 0

if dim M = 2k + 1 then ind(D) = 0

Remark: the index is a very stable object


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Examples.

The Gauss-Bonnet operator d + d∗

with E = ΛevevM ⊕ ΛoddM;

the spin-Dirac operator Dspin ≡ D/ on a spin manifoldwith E = S/ = S/+ ⊕ S/− the spinor bundle;

the signature operator on an orientable manifold Dsign

with E = Λ+M ⊕ Λ−M defined in terms of Hodge-?;

the Dolbeault operator ∂ + ∂∗.

with E = Λ0,evevM ⊕ Λ0,oddM


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Atiyah-Singer index theory.

Atiyah-Singer index formula

ind D+ =

∫M

AS(RM ,RE ) = < [AS(RM ,RE ), [M] >

Right hand side is topological and often even homotopical

Geometric applications for Gauss-Bonnet, signature and Dolbeault:first prove by Hodge-de Rham-Dolbeault that

χ(M) = ind(d + d∗)+; sign(M) = ind D+,sign; χ(M,O) = ind(∂ + ∂∗)+

then apply Atiyah-Singer and get Chern-Gauss-Bonnet, Hirzebruchand Riemann-Roch:

χ(M) =

∫M

Pf(M); sign(M) =

∫M

L(M); χ(M,O) =

∫M

Td(M)


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More geometric applications.

Assume that M4k is spin; then ind D+,spin =∫M A(M)

if g is of positive scalar curvature then Dspin is invertible because ofLichnerowicz formula

it follows that the topological term∫M A(M) must be zero

⇒ obstructions to existence of positive scalar curvature metrics.


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More about the index on compact manifolds without boundary

the index depends only on 0-eigenvalue

index is a bordism invariant (if M is a boundary than ind D+ = 0).

ind D+ ≡ TrΠ+ − TrΠ− = Tr(S+)− Tr(S−) where S± ∈ Ψ−∞ areremainders in a parametrix construction

Here Π± are the orthogonal projections onto the kernel of D±.

(Parametrix: an operator Q : C∞(M,E−)→ C∞(M,E+) which is aninverse of D+ modulo smoothing operators: D+Q = Id + S−;QD+ = Id + S+.)

parametrices and remainders S± can be localized near the diagonal

⇒ index data are ”localized near the diagonal”

very special of the index; more sophisticated spectral invariant cannotbe localized.


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Eta invariants

what about others spectral invariants ?

the eta invariant is a fundamental example; let us see the definition

(M, g) is a now odd dimensional

the eta invariant associated to a Dirac operator D is by definition

η(D) :=2√π

∫ ∞0

Tr(D exp(−(tD)2)dt

η(D) is the value at s = 0 of the meromorphic continuation of∑λ6=0 sign(λ)|λ|−s Res >> 0.

η(D) measures the spectral asymmetry of the self-adjont op. D.

η(D) is a very sensitive invariant.

Indeed, if {Dt} is a one-parametr family of operators then (assumingfor simplicity D0 and D1 invertible)

η(D1)− η(D0) =

∫M

local + SF({Dt})

rho-invariants (defined next) are more stable objects.


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The Atiyah-Patodi-Singer index theorem

where does eta come from ?

η is the boundary correction term in the index theorem on manifoldswith boundary:

Atiyah-Patodi-Singer index theorem: on an even dimensionalmanifold W with boundary equal to M and metric G of product typenear the boundary:

indAPS (D+W ) =

∫AS − η(D) + dim(Ker(D))

2

where AS is the Atiyah-Singer integrand.

remark: this index is defined by a boundary value problem

equivalently we can look at the manifold with cylindrical end definedby the manifold with boundary and take the L2-index


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Atiyah-Patodi-Singer rho invariant

it is associated to the choice of a pair of finite dimensional unitaryrepresentations of π1(M) := Γ of the same dimension:λ1, λ2 : Γ→ U(CN) .we consider Lj := M ×λj

CN (a flat bundle endowed with a naturalunitary connection).we can twist D by Lj obtaining two operators DL1 and DL2 .then the Atiyah-Patodi-Singer rho invariant is by definition

ρ(D)λ1−λ2 := η(DL1)− η(DL2)

this is a more stable invariant than eta itselfparticularly useful when π1(M) is a torsion groupfor example: in distinguishing metrics of Positive Scalar Curvature(PSC)for example: in distinguishing the diffeomorphism type ofhomotopically equivalent manifoldsthe rho-invariant is a secondary invariant (e.g.: the index for apositive scalar curvature metric is zero but rho is not)


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A hierarchy of geometric structures

There is a hierarchy of geometric structures in index theory:

1 a compact manifold M,

2 a fibration X → B with fiber M; for example M × B → B

3 a Galois Γ-coverings M → M, for example the universal cover of M(then Γ = π1(M))

4 a measured foliation

5 a general foliation.


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Fundamental example.

take a Γ-covering M → M,

take a Γ-manifold T ,

consider the product fibration M × T → T ;

consider the quotient X := (M × T )/Γ by the diagonal action.

X is foliated by the images of the fibers of M × T → T

get a foliation (X ,F)


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This example presents all the structures in the hierarchy (depending on Tand Γ):

If T = point and Γ = {1} we have a compact manifold.

If Γ = {1} we simply have a fibration.

If T = point but Γ 6= {1} then we have a Galois covering.

If dim T > 0, Γ 6= {1}, and if T has a Γ-invariant measure, then wehave a measured foliation

If dim T > 0, Γ 6= {1}, then we have a general foliation.


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Specific examples of foliations

Example

M = R, the universal cover of M := S1; T = S1,

Γ = Z,

action on R× S1 given by

n · (r , e iθ) =(r + n, e i(θ + nα)

)for some α ∈ R

Then X = T 2 and if α ∈ R \Q we get the Kronecker foliation.

this is a measured foliation


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Example

consider a smooth closed riemann surface Σ of genus g > 1 and letΓ = π1(Σ), a discrete subgroup of PSL(2,R).

Consider the universal cover H2 → Σ,

Consider T = S1 and Γ acting on S1 by fractional lineartransformations.

Then we getH2 × S1 → S1

and the quotient X := (H2 × S1)/Γ is a 3 manifold with a highlynon-trivial foliation.

it can be proved that this foliation is not measured.


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A hierarchy of Dirac operators

On each geometric structure we consider the natural notion of Diracoperator:

a classical Dirac operator if Γ = {1}, T =point (compact manifold);

a smooth family of Dirac operators if Γ = {1} but dim T > 0; (in thiscase we have a trivial fibration M ×T → T ); more generally we couldconsider a smooth non-trivial fibration X → T and a vertical family(Dθ)θ∈T ;

a Γ-invariant operator D on a Galois covering M → M if Γ 6= {1} butT = point;

a Γ-equivariant family (Dθ)θ∈T in the general case of a Γ-fibrationM × T → T ;notice that in this case we get a longitudinal Diracoperator on the foliation (X ,F) with X = (M × T )/Γ


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A hierarchy of NUMERIC invariantsWe now assume that our Γ-equivariant fibration is such that T admits aΓ-invariant measure ν.This is a non-trivial assumption.

M X → T M X = (M × T )/Γ

Dirac D (Dθ)θ∈T D (Γ-invariant) (Dθ)θ∈T (Γ-equiv.)

Index Ind D∫T Ind Dθdθ Ind(2) D Indν(Dθ)

theorems AS AS Atiyah Connes

eta η(D)∫B η(Dθ)dθ η(2)(D) ην(Dθ)

rho APS APS Cheeger-Gromov Benameur-P.

if ∂( ) 6= ∅ APS APS Ramachandran Ramachandran

The numerical indeces of Atiyah and Connes are obtained by definingsuitable Von Neumann algebras with traces, proving that the projectionsonto the kernel of D+ and D− are elements of this Von Neumann algebras(and that they have finite traces) and then taking the differences of thesetraces.


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HIGHER index theory

(We do not assume anymore that T admits a Γ-invariant measure.)

First of all, what is higher index theory ?

Answer:I define an index as a class in the K-theory group of a suitable

C∗-algebra;I extract numerical invariants out of this index class (higher indeces);I prove geometrical formulas for these higher indeces (in the spirit of

Atiyah-Singer)I study stability properties of the higher indeces and obtain in this way

information about the geometric invariants appearing on the right handside of the index formulas


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HIGHER index theory: current situation

X → T M X = (M × T )/Γ

Dirac (Dθ)θ∈T D (Γ-invariant) (Dθ) (Γ-equiv.)

Index class ∈ K∗(C (T )) ∈ K∗(C∗r Γ) ∈ K∗(C (T ) or Γ)

theoremsAS

Bismut

Connes-Moscovici

Lott

ConnesMoriyoshi-NatsumeGorokowsky-Lott

higher eta Bismut-Cheeger LottLeichtnam-P.if Γ polyn. growthMoriyoshi-P.

higher rho Azzali ?? (few examples) ??

if ∂( ) 6= ∅ Bismut-CheegerMelrose-P.

Leichtnam-P.Leichtnam-P.if Γ polyn. growthMoriyoshi-P.

Everything in red or in magenta involves Getzler rescaling, heat calculus...


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ReferencesFor numerical invariants one can consult the paper:Moulay Benameur and Paolo Piazza. Index, eta and rho invariants onfoliated bundles.Asterisque vol 327, p. 201-287, 2009.

For higher index theory on fibrations and Galois coverings on manifoldswith or without boundary one can consult the long survey:Eric Leichtnam and Paolo Piazza. Elliptic operators and higher signatures.Ann. Inst. Fourier vol. 54 (2004) pp. 1197-1277

For higher index theory on foliations without boundary one should look atthe seminal papers of Connes-Skandalis and Connes:Alain Connes and George Skandalis. The longitudinal index theorem forfoliations. Publ. Res. Inst. Math. Sci., 20(6):11391183, 1984.Alain Connes. Cyclic cohomology and the transverse fundamental class ofa foliation. In Geometric methods in operator algebras (Kyoto, 1983),volume 123 of Pitman Res. Notes Math. Ser., pages 52144. Longman Sci.Tech., Harlow, 1986.


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A very well written paper for a special case of the results established byConnes in this last article (the so called Godbillon-Vey index theorem) is:Hitoshi Moriyoshi and Toshikazu Natsume. The Godbillon-Vey cycliccocycle and longitudinal Dirac operators. Pacific J. Math., 172(2):483539,1996.The case with boundary is treated in the recent paper:Hitoshi Moriyoshi and Paolo Piazza. Eta cocycles, relative pairings and theGodbillon-Vey index theorem. Preprint February 2011. (arXiv:0907.0173).107 pp. To appear in GAFA .


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higher index theory: a survey. · plan of the talk: 1 dirac operators 2 atiyah-singer index theory...

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