Higher index theory: a survey. Paolo Piazza (Sapienza Universit` a di Roma) Incontri di geometria noncommutativa Napoli, Settembre 2012. Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 Incontri di geometria noncommutativa Napoli, Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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) Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 coker P = dim ker P − dim ker P ∗ if dim M = 2 k then E is graded, E = E + ⊕ E − and D is odd: � � D − 0 D − = ( D + ) ∗ D = D + 0 if dim M = 2 k , ind( D ) = 0 (since D = D ∗ ), but ind D + � = 0 if dim M = 2 k + 1 then ind( D ) = 0 Remark: the index is a very stable object Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
Examples. The Gauss-Bonnet operator d + d ∗ with E = Λ evev M ⊕ Λ odd M ; the spin-Dirac operator D spin ≡ D / on a spin manifold / + ⊕ S / − the spinor bundle; with E = S / = S the signature operator on an orientable manifold D sign with E = Λ + M ⊕ Λ − M defined in terms of Hodge- ⋆ ; ∗ . the Dolbeault operator ∂ + ∂ with E = Λ 0 , evev M ⊕ Λ 0 , odd M Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
Atiyah-Singer index theory. Atiyah-Singer index formula � ind D + = AS ( R M , R E ) = < [ AS ( R M , R E ) , [ M ] > 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 ) = Pf ( M ); sign ( M ) = L ( M ); χ ( M , O ) = Td ( M ) M M M Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
More geometric applications. � Assume that M 4 k is spin; then ind D + , spin = M � A ( M ) if g is of positive scalar curvature then D spin is invertible because of Lichnerowicz formula � M � it follows that the topological term A ( M ) must be zero ⇒ obstructions to existence of positive scalar curvature metrics. Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 + ≡ Tr Π + − Tr Π − = 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. Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 � ∞ 2 Tr( D exp( − ( tD ) 2 ) dt η ( D ) := √ π 0 η ( D ) is the value at s = 0 of the meromorphic continuation of � λ � =0 sign ( λ ) | λ | − s Re s >> 0 . η ( D ) measures the spectral asymmetry of the self-adjont op. D . η ( D ) is a very sensitive invariant. Indeed, if { D t } is a one-parametr family of operators then (assuming for simplicity D 0 and D 1 invertible) � η ( D 1 ) − η ( D 0 ) = local + SF ( { D t } ) M rho-invariants (defined next) are more stable objects. Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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: � AS − η ( D ) + dim(Ker( D )) ind APS ( D + W ) = 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 L 2 -index Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
Atiyah-Patodi-Singer rho invariant it is associated to the choice of a pair of finite dimensional unitary representations of π 1 ( M ) := Γ of the same dimension: λ 1 , λ 2 : Γ → U ( C N ) . M × λ j C N (a flat bundle endowed with a natural we consider L j := � unitary connection). we can twist D by L j obtaining two operators D L 1 and D L 2 . then the Atiyah-Patodi-Singer rho invariant is by definition ρ ( D ) λ 1 − λ 2 := η ( D L 1 ) − η ( D L 2 ) this is a more stable invariant than eta itself particularly useful when π 1 ( 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) Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 = π 1 ( M )) 4 a measured foliation 5 a general foliation. Incontri di geometria noncommutativa Napoli, Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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 ) Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 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. Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
Specific examples of foliations Example � M = R , the universal cover of M := S 1 ; T = S 1 , Γ = Z , action on R × S 1 given by � r + n , e i ( θ + n α ) � n · ( r , e i θ ) = for some α ∈ R Then X = T 2 and if α ∈ R \ Q we get the Kronecker foliation. this is a measured foliation Incontri di geometria noncommutativa Napol Paolo Piazza (Sapienza Universit` a di Roma) () Higher index theory: a survey. / 22
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