Higher index theory: a survey. Paolo Piazza (Sapienza Universit` a - - PowerPoint PPT Presentation

Higher index theory: a survey.

Paolo Piazza (Sapienza Universit` a di Roma) Incontri di geometria noncommutativa Napoli, Settembre 2012.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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 Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napoli, / 22

Dirac-type operators. We consider a riemannian manifold (M, g) without boundary and a Dirac 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 complex bundle 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 E to be metric-compatible ⇒ D = D∗ (examples in a moment)

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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 =

• D−

D+

• D− = (D+)∗

if dim M = 2k, ind(D) = 0 (since D = D∗), but ind D+ = 0 if dim M = 2k + 1 then ind(D) = 0 Remark: the index is a very stable object

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

Examples. The Gauss-Bonnet operator d + d∗ with E = ΛevevM ⊕ ΛoddM; the spin-Dirac operator Dspin ≡ D / on a spin manifold with 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

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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, Hirzebruch and Riemann-Roch: χ(M) =

• M

Pf(M); sign(M) =

• M

L(M); χ(M, O) =

• M

Td(M)

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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 of Lichnerowicz formula it follows that the topological term

• M

A(M) must be zero ⇒ obstructions to existence of positive scalar curvature metrics.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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−∞ are remainders 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 an inverse 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 cannot be localized.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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 ∞ Tr(D exp(−(tD)2)dt η(D) is the value at s = 0 of the meromorphic continuation of

• λ=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 (assuming for simplicity D0 and D1 invertible) η(D1) − η(D0) =

• M

local + SF({Dt}) rho-invariants (defined next) are more stable objects.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

The Atiyah-Patodi-Singer index theorem where does eta come from ? η is the boundary correction term in the index theorem on manifolds with boundary: Atiyah-Patodi-Singer index theorem: on an even dimensional manifold W with boundary equal to M and metric G of product type near 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 defined by the manifold with boundary and take the L2-index

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

Atiyah-Patodi-Singer rho invariant it is associated to the choice of a pair of finite dimensional unitary representations 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 natural unitary 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 itself particularly useful when π1(M) is a torsion group for example: in distinguishing metrics of Positive Scalar Curvature (PSC) for example: in distinguishing the diffeomorphism type of homotopically equivalent manifolds the rho-invariant is a secondary invariant (e.g.: the index for a positive scalar curvature metric is zero but rho is not)

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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. Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napoli, / 22

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)

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

This example presents all the structures in the hierarchy (depending on T and 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 = {1} then we have a Galois covering. If dim T > 0, Γ = {1}, and if T has a Γ-invariant measure, then we have a measured foliation If dim T > 0, Γ = {1}, then we have a general foliation.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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, eiθ) =

• r + n, ei(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

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

Example

consider a smooth closed riemann surface Σ of genus g > 1 and let Γ = 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 linear transformations. Then we get H2 × S1 → S1 and the quotient X := (H2 × S1)/Γ is a 3 manifold with a highly non-trivial foliation. it can be proved that this foliation is not measured.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

A hierarchy of Dirac operators

On each geometric structure we consider the natural notion of Dirac

• perator:

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 this case we have a trivial fibration M × T → T); more generally we could consider 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 = {1} but T = point; a Γ-equivariant family ( Dθ)θ∈T in the general case of a Γ-fibration

• M × T → T;notice that in this case we get a longitudinal Dirac
• perator on the foliation (X, F) with X = (

M × T)/Γ

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

A hierarchy of NUMERIC invariants

We now assume that our Γ-equivariant fibration is such that T admits a Γ-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 ∂( ) = ∅ APS APS Ramachandran Ramachandran The numerical indeces of Atiyah and Connes are obtained by defining suitable Von Neumann algebras with traces, proving that the projections

• nto 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 these traces.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

HIGHER index theory

(We do not assume anymore that T admits a Γ-invariant measure.) First of all, what is higher index theory ? Answer:

◮ define an index as a class in the K-theory group of a suitable

C ∗-algebra;

◮ extract numerical invariants out of this index class (higher indeces); ◮ prove geometrical formulas for these higher indeces (in the spirit of

Atiyah-Singer)

◮ study stability properties of the higher indeces and obtain in this way

information about the geometric invariants appearing on the right hand side of the index formulas

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

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) ⋊r Γ) theorems AS Bismut Connes

• Moscovici

Lott Connes Moriyoshi-Natsume Gorokowsky-Lott higher eta Bismut-Cheeger Lott Leichtnam-P. if Γ polyn. growth Moriyoshi-P. higher rho Azzali ?? (few examples) ?? if ∂( ) = ∅ Bismut-Cheeger Melrose-P. Leichtnam-P. Leichtnam-P. if Γ polyn. growth Moriyoshi-P. Everything in red or in magenta involves Getzler rescaling, heat calculus...

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

References

For numerical invariants one can consult the paper: Moulay Benameur and Paolo Piazza. Index, eta and rho invariants on foliated bundles. Asterisque vol 327, p. 201-287, 2009. For higher index theory on fibrations and Galois coverings on manifolds with 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 at the seminal papers of Connes-Skandalis and Connes: Alain Connes and George Skandalis. The longitudinal index theorem for

• foliations. Publ. Res. Inst. Math. Sci., 20(6):11391183, 1984.

Alain Connes. Cyclic cohomology and the transverse fundamental class of a foliation. In Geometric methods in operator algebras (Kyoto, 1983), volume 123 of Pitman Res. Notes Math. Ser., pages 52144. Longman Sci. Tech., Harlow, 1986.

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22

A very well written paper for a special case of the results established by Connes in this last article (the so called Godbillon-Vey index theorem) is: Hitoshi Moriyoshi and Toshikazu Natsume. The Godbillon-Vey cyclic cocycle 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 the Godbillon-Vey index theorem. Preprint February 2011. (arXiv:0907.0173). 107 pp. To appear in GAFA .

Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. Incontri di geometria noncommutativa Napol / 22