Invariants from noncommutative index theory for homotopy equivalences

Transcription

1 Invariants from noncommutative index theory for homotopy equivalences Charlotte Wahl ECOAS 2010 Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

2 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 functional calculus, dense, etc.) A = C (B) Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

3 Basics in noncommutative index theory: A noncommutative Chern character Given one gets A C -algebra p. e. A = C(B), B closed manifold A A smooth subalgebra (=closed under holomorphic functional calculus, dense, etc.) A = C (B) a Z-graded Fréchet algebra ˆΩ A of noncommutative differential forms with differential d : ˆΩ k A ˆΩ 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) Invariants for homotopy equivalences ECOAS / 12

4 Basics in noncommutative index theory: A noncommutative Chern character Given one gets A C -algebra p. e. A = C(B), B closed manifold A A smooth subalgebra (=closed under holomorphic functional calculus, dense, etc.) A = C (B) a Z-graded Fréchet algebra ˆΩ A of noncommutative differential forms with differential d : ˆΩ k A ˆΩ 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) Invariants for homotopy equivalences ECOAS / 12

5 Dirac operators over C -algebras Given (M, g) closed oriented Riemannian manifold E M hermitian bundle with Clifford action and compatible connection ( 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) Invariants for homotopy equivalences ECOAS / 12

6 Dirac operators over C -algebras Given (M, g) closed oriented Riemannian manifold E M hermitian bundle with Clifford action and compatible connection ( 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 Mishenko-Fomenko 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) Invariants for homotopy equivalences ECOAS / 12

7 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) Invariants for homotopy equivalences ECOAS / 12

8 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) ch(ind(d F )) = Â(M) ch(e/s) ch(f) M H dr (A ). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

9 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) ch(ind(d F )) = Â(M) ch(e/s) ch(f) M H dr (A ). 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) Invariants for homotopy equivalences ECOAS / 12

10 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) ˆΩ A /[ˆΩ A, ˆΩ A ] + d ˆΩ A generalizing the classical η-invariant (with A = C) η(d F, A) = 1 t 1 2 Tr(DF e t(d F +A) 2 )dt. π 0 Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

11 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) ˆΩ A /[ˆΩ A, ˆΩ A ] + d ˆΩ A generalizing the classical η-invariant (with A = C) η(d F, A) = 1 t 1 2 Tr(DF e t(d F +A) 2 )dt. π 0 Higher η-invariants were introduced by Lott (1992). The general definition is implicit in work of Lott (1999). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

12 Atiyah-Patodi-Singer index theorem Let M be an oriented Riemannian manifold with cylindric end Z = IR + M. On Z all structures are assumed of product type. 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 DF + A is invertible. Let χ : M IR be smooth, supp χ Z; supp(χ 1) compact. Proposition (W., 2009) ch ind(d + P c(dx)χ(x)a) = M Â(M) ch(e/s) ch(f) η(d F, A). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

13 Atiyah-Patodi-Singer index theorem Let M be an oriented Riemannian manifold with cylindric end Z = IR + M. On Z all structures are assumed of product type. 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 DF + A is invertible. Let χ : M IR be smooth, supp χ Z; supp(χ 1) compact. Proposition (W., 2009) ch ind(d + P c(dx)χ(x)a) = M Â(M) ch(e/s) ch(f) η(d F, A). The proposition generalizes the higher APS index theorem proven by Leichtnam-Piazza ( ). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

14 Higher ρ-invariants for homotopy equivalences Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M N a smooth orientation preserving homotopy equivalence. A = C Γ, Γ = π 1 (N). F N = Ñ Γ C Γ Mishenko-Fomenko bundle, F M = f F N. D F signature operator on N M op twisted by F N F M. Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

15 Higher ρ-invariants for homotopy equivalences Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M N a smooth orientation preserving homotopy equivalence. A = C Γ, Γ = π 1 (N). F N = Ñ Γ C Γ Mishenko-Fomenko bundle, F M = f F N. D F signature operator on N M op twisted by F N F M. Piazza-Schick (2007) constructed a smoothing operator A such that D F + A is invertible (based on results of Hilsum-Skandalis 1992). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

16 Higher ρ-invariants for homotopy equivalences Let M, N be odd-dimensional oriented closed Riemannian manifolds, f : M N a smooth orientation preserving homotopy equivalence. A = C Γ, Γ = π 1 (N). F N = Ñ Γ C Γ Mishenko-Fomenko bundle, F M = f F N. D F signature operator on N M op twisted by F N F M. Piazza-Schick (2007) constructed a smoothing operator A such that D F + A is invertible (based on results of Hilsum-Skandalis 1992). Let ˆΩ <e> (A ) = C < g 0 d g 1... d g m g 0 g 1... g m = e > ˆΩ A. Definition ρ(f ) := [η(d F, A)] ˆΩ A /[ˆΩ A, ˆΩ A ] + d ˆΩ A + ˆΩ <e> A. Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

17 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

18 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

19 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

20 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. 4 ρ is well-defined on the surgery structure set S(N). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

21 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. 4 ρ is well-defined on the surgery structure set S(N). 5 By pairing with suitable traces one recovers ρ APS (N) ρ APS (M) resp. ρ L 2(N) ρ L 2(M) (Piazza-Schick 2007). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

22 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. 4 ρ is well-defined on the surgery structure set S(N). 5 By pairing with suitable traces one recovers ρ APS (N) ρ APS (M) resp. ρ L 2(N) ρ L 2(M) (Piazza-Schick 2007). 6 (Product formula) If N = N 1 X, M = M 1 X, f = f 1 id X, then ρ(f ) = ρ(f 1 ) ch(ind(d FX )). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

23 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. 4 ρ is well-defined on the surgery structure set S(N). 5 By pairing with suitable traces one recovers ρ APS (N) ρ APS (M) resp. ρ L 2(N) ρ L 2(M) (Piazza-Schick 2007). 6 (Product formula) If N = N 1 X, M = M 1 X, f = f 1 id X, then ρ(f ) = ρ(f 1 ) ch(ind(d FX )). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

24 Properties 1 ρ(f ) does not depend on the choices involved in the definition of A. 2 ρ(f ) is independent of the metrics on M and N. 3 ρ(f ) depends on f only up to homotopy. 4 ρ is well-defined on the surgery structure set S(N). 5 By pairing with suitable traces one recovers ρ APS (N) ρ APS (M) resp. ρ L 2(N) ρ L 2(M) (Piazza-Schick 2007). 6 (Product formula) If N = N 1 X, M = M 1 X, f = f 1 id X, then ρ(f ) = ρ(f 1 ) ch(ind(d FX )). The proofs use the APS index theorem. Local terms vanish since we divided out ˆΩ <e> A. (4) is based on a generalization of results of Hilsum-Skandalis (1992) to manifolds with cylindric ends. (6) uses a product formula for noncommutative η-forms (W., 2009). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

25 Applications dim N = 4k 1, k 2, Γ = π 1 (N) not torsion free Proposition (Chang-Weinberger 2003) There are homotopy equivalences f i : M i N, i IN such that ρ L 2(M i ) ρ L 2(M j ), i j. Thus [(M i, f i )] are distinct in S(N). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

26 Applications dim N = 4k 1, k 2, Γ = π 1 (N) not torsion free Proposition (Chang-Weinberger 2003) There are homotopy equivalences f i : M i N, i IN such that ρ L 2(M i ) ρ L 2(M j ), i j. Thus [(M i, f i )] are distinct in S(N). Corollary Let X be a closed manifold with a non-zero higher signature. Assume that π 1 (X ), Γ are Gromov hyperbolic. Then [(M i X, f i id X )] are distinct in S(N X ) and distinguished by ρ(f i id X ). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

27 Applications dim N = 4k 1, k 2, Γ = π 1 (N) not torsion free Proposition (Chang-Weinberger 2003) There are homotopy equivalences f i : M i N, i IN such that ρ L 2(M i ) ρ L 2(M j ), i j. Thus [(M i, f i )] are distinct in S(N). Corollary Let X be a closed manifold with a non-zero higher signature. Assume that π 1 (X ), Γ are Gromov hyperbolic. Then [(M i X, f i id X )] are distinct in S(N X ) and distinguished by ρ(f i id X ). Idea of proof: A non-zero higher signature implies that ch ind(d FX ) 0. Now apply product formula to ρ(f i id). Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

28 Open questions I Γ = π 1 (N) Does the following diagram commute? L n+1 ( ZΓ) K n+1 (C Γ) ch S(N) ˆΩ A /[ˆΩ A, ˆΩ A ] + d ˆΩ A + ˆΩ <e> A. A positive answer to this question would lead to more general applications. ρ Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

29 Open questions II What is the connection with the Higson-Roe map from surgery to analysis (2004)? L n+1 ( ZΓ) S(N) N (N) L n ( ZΓ) K n+1 (C r Γ) K n+1 (D Γ N) K n (BΓ) K n (C r Γ) Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

30 Open questions II What is the connection with the Higson-Roe map from surgery to analysis (2004)? L n+1 ( ZΓ) S(N) N (N) L n ( ZΓ) K n+1 (C r Γ) K n+1 (D Γ N) K n (BΓ) K n (C r Γ) the Higson-Roe interpretation of the APS ρ-invariant (2010)? K n+1 (C Γ) lim K n+1 (DΓ X ) X BΓ ρ APS K n (BΓ) Z IR IR/ Z Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12

31 Some literature S. Chang & S. Weinberger, On invariants of Hirzebruch and Cheeger-Gromov, Geom. Topol. 7 (2003), pp N. Higson & J. Roe, Mapping surgery to analysis. I-III. K-theory 33 (2004), pp P. Piazza & T. Schick, Bordism, rho-invariants and the Baum-Connes conjecture, J. Noncommut. Geom. 1 (2007), pp N. Higson & J. Roe, K-homology, assembly and rigidity theorems for relative eta invariants, Pure Appl. Math. Quart. 6 (2010), pp C. W., Higher ρ-invariants and the surgery structure set, arxiv: Charlotte Wahl (Hannover) Invariants for homotopy equivalences ECOAS / 12