Higher rho-invariants for the signature operator A survey and - PowerPoint PPT Presentation

Higher rho-invariants for the signature operator A survey and perspectives Charlotte Wahl Hannover Copenhagen, 11-15/6/2018 Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 1 / 20

Basics in noncommutative index theory: A noncommutative Chern character Given A C ∗ -algebra p. e. A = C ( B ), B closed manifold A ∞ ⊂ A “smooth” subalgebra (=closed under holomorphic A ∞ = C ∞ ( B ) functional calculus, dense, etc.) Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Basics in noncommutative index theory: A noncommutative Chern character Given A C ∗ -algebra p. e. A = C ( B ), B closed manifold A ∞ ⊂ A “smooth” subalgebra (=closed under holomorphic A ∞ = C ∞ ( B ) functional calculus, dense, etc.) one gets echet algebra ˆ Ω ∗ A ∞ of noncommutative differential a Z Z-graded Fr´ forms with differential d : ˆ Ω k A ∞ → ˆ Ω k +1 A ∞ a Chern character ch : K ∗ ( A ) → H dR ∗ ( A ∞ ) H dR ∗ ( A ∞ ) pairs with continuous reduced cyclic cocycles on A ∞ Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Basics in noncommutative index theory: A noncommutative Chern character Given A C ∗ -algebra p. e. A = C ( B ), B closed manifold A ∞ ⊂ A “smooth” subalgebra (=closed under holomorphic A ∞ = C ∞ ( B ) functional calculus, dense, etc.) one gets echet algebra ˆ Ω ∗ A ∞ of noncommutative differential a Z Z-graded Fr´ forms with differential d : ˆ Ω k A ∞ → ˆ Ω k +1 A ∞ a Chern character ch : K ∗ ( A ) → H dR ∗ ( A ∞ ) H dR ∗ ( A ∞ ) pairs with continuous reduced cyclic cocycles on A ∞ Motivating example used in higher index theory: Γ finitely generated group with length function, A = C ∗ Γ, A ∞ the Connes-Moscovici algebra Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 2 / 20

Dirac operators over C ∗ -algebras Given ( M , g ) closed oriented Riemannian manifold E → M hermitian bundle with Clifford action and compatible connection (Z Z / 2-graded, if dim M even) P ∈ C ∞ ( M , M n ( A ∞ )) projection we get an A -vector bundle F := P ( A n × M ) → M and a (odd) Dirac operator D F : C ∞ ( M , E ⊗ F ) → C ∞ ( M , E ⊗ F ). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 3 / 20

Dirac operators over C ∗ -algebras Given ( M , g ) closed oriented Riemannian manifold E → M hermitian bundle with Clifford action and compatible connection (Z Z / 2-graded, if dim M even) P ∈ C ∞ ( M , M n ( A ∞ )) projection we get an A -vector bundle F := P ( A n × M ) → M and a (odd) Dirac operator D F : C ∞ ( M , E ⊗ F ) → C ∞ ( M , E ⊗ F ). Important example for higher index theory: the Mishchenko C ∗ Γ-vector bundle: F = ˜ M × Γ C ∗ Γ with Γ = π 1 ( M ). E = S the spinor bundle (gives twisted spin Dirac operator) E = Λ ∗ ( T ∗ M ) (gives twisted de Rham or signature operator). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 3 / 20

Index theory The Dirac operator D F is Fredholm on the Hilbert A -module L 2 ( M , E ⊗ F ) with ind ( D F ) ∈ K ∗ ( A ). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Index theory The Dirac operator D F is Fredholm on the Hilbert A -module L 2 ( M , E ⊗ F ) with ind ( D F ) ∈ K ∗ ( A ). Proposition (Atiyah-Singer index theorem) � ˆ ∈ H dR ch( ind ( D F )) = A ( M ) ch( E / S ) ch( F ) ∗ ( A ∞ ) . M Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Index theory The Dirac operator D F is Fredholm on the Hilbert A -module L 2 ( M , E ⊗ F ) with ind ( D F ) ∈ K ∗ ( A ). Proposition (Atiyah-Singer index theorem) � ˆ ∈ H dR ch( ind ( D F )) = A ( M ) ch( E / S ) ch( F ) ∗ ( A ∞ ) . M Application in higher index theory: If D F is the signature operator twisted by F = ˜ M × Γ C ∗ Γ , then ind ( D F ) is homotopy invariant. The proposition implies: By pairing ch( ind ( D F )) with cyclic cocycles one gets higher signatures. This can be used to prove the Novikov conjecture for Gromov hyperbolic groups (Connes–Moscovici 1990). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 4 / 20

Secondary invariants Let A be a smoothing symmetric operator on L 2 ( M , E ⊗ F ) such that D F + A is invertible. ( A should be odd if dim M is even.) Then one can define η ( D F , A ) ∈ ˆ Ω ∗ A ∞ / [ˆ Ω ∗ A ∞ , ˆ Ω ∗ A ∞ ] + d ˆ Ω ∗ A ∞ generalizing the classical η -invariant (with A = C ) � ∞ 1 t − 1 2 Tr ( D F e − t ( D F + A ) 2 ) dt . √ π η ( D F , A ) = 0 Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 5 / 20

Secondary invariants Let A be a smoothing symmetric operator on L 2 ( M , E ⊗ F ) such that D F + A is invertible. ( A should be odd if dim M is even.) Then one can define η ( D F , A ) ∈ ˆ Ω ∗ A ∞ / [ˆ Ω ∗ A ∞ , ˆ Ω ∗ A ∞ ] + d ˆ Ω ∗ A ∞ generalizing the classical η -invariant (with A = C ) � ∞ 1 t − 1 2 Tr ( D F e − t ( D F + A ) 2 ) dt . √ π η ( D F , A ) = 0 Higher η -invariants were introduced by Lott (1992). The general definition is implicit in work of Lott (1999). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 5 / 20

Atiyah–Patodi–Singer index theorem Let M be an oriented Riemannian manifold with cylindric end R + × ∂ M . On Z all structures are assumed of product type. Z = I If dim M is even, then on Z D + F = c ( dx )( ∂ x − D ∂ F ) . Let A be a symmetric, smoothing operator on L 2 ( ∂ M , E + ⊗ F ) such that D ∂ F + A is invertible. Let χ : M → I R be smooth, supp χ ⊂ Z ; supp( χ − 1) compact. Proposition (W., 2009) � ch ind ( D + A ( M ) ch( E / S ) ch( F ) − η ( D ∂ ˆ P − c ( dx ) χ ( x ) A ) = F , A ) . M The proposition generalizes the higher APS index theorem proven by Leichtnam–Piazza (1997–2000). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 6 / 20

Higher ρ -invariants for the signature operator (Leichtnam–Piazza, Wahl) M oriented Riemannian manifold with dim M = 2 m − 1. Assume that the m -th Novikov Shubin invariant of M is ∞ + . Let D F be the signature operator twisted by the Mishchenko bundle F . Let I be the involution which is 1 on forms of degree < m and − 1 on the complement. Then for t small D F + tI is invertible. We define Ω � e � ρ ( M ) := [ η ( D F , tI )] ∈ ˆ Ω ∗ A ∞ / [ˆ Ω ∗ A ∞ , ˆ Ω ∗ A ∞ ] + d ˆ Ω ∗ A ∞ + ˆ ∗ A ∞ . Product formula: If both sides are defined, then ρ ( M × N ) = ρ ( M ) ch(sign( N )) . Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 7 / 20

Generalization to an almost flat setting I (Azzali–W., work in progress) Given c ∈ H 2 (Γ) one may construct a Mishchenko bundle F σ s associated to a twisted group C ∗ -algebra C ∗ (Γ , σ s ) with [ σ s ] = e isc ∈ H 2 (Γ , U (1)). The bundle is almost flat. The C ∗ -algebras assemble to an upper-semicontinuous field. Crucial property: If a differential operator associated to that field is invertible at s = 0, then it is invertible for s near 0. In this case we can guarantee that the signature operator twisted by F σ s can be perturbed to an invertible operator and define the higher ρ -invariants. They have the usual properties: Metric independence, product formula. Some properties follow from the flat case using the Hanke–Schick method of comparison. Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 8 / 20

Generalization to an almost flat setting II ( F n , ∇ n ) sequence of Hermitian bundles with connection such that ( ∇ n ) 2 n →∞ → 0. Can we construct an invertible perturbation for the signature operator such that it leads to a well-defined ρ -invariant? Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 9 / 20

Generalization to an almost flat setting III For geometric applications, one wants to assign a ρ -invariant to a map f : M → B Γ, thus better start with quasirepresentations of Γ instead of flat bundles: Fix a finite set F ⊂ Γ and ε > 0. A map φ : Γ → A is an ( F , ε ) -unitary representation if φ ( e ) = 1 and it holds that � φ ( g ) � ≤ 1, ∀ g ∈ Γ , � φ ( g − 1 ) − φ ( g ) ∗ � ≤ ε ∀ g ∈ F , � φ ( gh ) − φ ( g ) φ ( h ) � ≤ ε ∀ g , h ∈ F . Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 10 / 20

Generalization to an almost flat setting III For geometric applications, one wants to assign a ρ -invariant to a map f : M → B Γ, thus better start with quasirepresentations of Γ instead of flat bundles: Fix a finite set F ⊂ Γ and ε > 0. A map φ : Γ → A is an ( F , ε ) -unitary representation if φ ( e ) = 1 and it holds that � φ ( g ) � ≤ 1, ∀ g ∈ Γ , � φ ( g − 1 ) − φ ( g ) ∗ � ≤ ε ∀ g ∈ F , � φ ( gh ) − φ ( g ) φ ( h ) � ≤ ε ∀ g , h ∈ F . For manifolds with psc metrics we define ρ -invariants for the spin Dirac operators twisted by almost flat bundles associated to a sequence of ( F n , 1 n )-unitary representations (see work by Dardalat). Charlotte Wahl (Hannover) Higher rho-invariants Copenhagen, 11-15/6/2018 10 / 20