Stratified surgery and the signature operator

Transcription

1 Stratified surgery and the signature operator Paolo Piazza (Sapienza Università di Roma). Index Theory and Singular structures. Toulouse, June 1st Based on joint work with Pierre Albin (and also Eric Leichtnam, Rafe Mazzeo and Thomas Schick).

2 Mapping surgery to analysis (Higson-Roe) I start by stating a fundamental theorem. Explanations in a moment. Theorem (N. Higson and J. Roe, 2004). Let V be a smooth, closed, oriented n-dimensional manifold and let Γ := π 1 (V ). We consider a portion of the surgery sequence in topology: L n+1 (ZΓ) S(V ) N (V ) L n (ZΓ). There are natural maps α, β, γ and a commutative diagram L n+1 (ZΓ) S(V ) N (V ) L n (ZΓ) α γ K n+1 (C (Ṽ )Γ ) K n+1 (D (Ṽ )Γ ) K n (V ) K n (C (Ṽ ) β The bottom sequence is the analytic surgery sequence associated to V and π 1 (V ). γ

3 Later Piazza-Schick gave a different description of the Higson-Roe theorem, employing Atiyah-Patodi-Singer index theory and using crucially the Hilsum-Skandalis perturbation associated to a homotopy equivalence. this more analytic treatment also gave the mapping of the Stolz surgery sequence for positive scalar curvature metrics to the same K-theory sequence.

4 The surgery sequence in topology the sequence actually extends to an infinite sequence to the left (but we only consider the displayed portion) L n+1 (ZΓ) S(V ) N (V ) L n (ZΓ). one of the goals of this sequence for V a manifold is to understand the structure set S(V ) S(V ) measures the non-rigidity of V (more later) L (ZΓ) are groups but S(V ) is only a set. N (V ) can be given the structure of a group but the map out of it is not a homomorphism. exactness must be suitably defined we now describe briefly the sequence

5 The structure set S(V ) and the normal set N (V ) Elements in S(V ) are equivalence classes [X f V ] with X smooth oriented and closed and f an orientation preserving homotopy equivalence. f (X 1 f 1 V ) 2 (X2 V ) if they are h-cobordant (there is a bordism X between X 1 and X 2 and a map F : X V [0, 1] such that F X1 = f 1 and F X2 = f 2 and F is a homotopy equivalence). S(V ) is a pointed set with [V Id V ] as a base point V is rigid if S(V ) = {[V Id V ]} N (V ) is the set of degree one normal maps f : M V considered up to normal bordism (we shall forget about the adjective normal in this talk) there is a natural map S(V ) N (V )

6 The L-groups. Exactness the L-groups L (ZΓ) are defined algebraically as equivalence classes of quadratic forms with coefficients in ZΓ a fundamental theorem of Wall tells us that L (ZΓ) is isomorphic to a bordism group L 1 (BΓ) of manifolds with b. In fact, one can choose yet a more specific realization with special cycles (L 2 (BΓ)); a special cycle is (W, W ) with a degree one normal map F : W V [0, 1] such that F W : W (V [0, 1]) is a homotopy equivalence + r : V BΓ through this special realization L n+1 (ZΓ) acts on S(V ) and exactness at S(V ) means the following: [X f V ] and [Y g V ] are mapped to the same element in N (V ) if and only if they belong to the same L n+1 (ZΓ)-orbit. the map N (V ) L n (ZΓ) is called the surgery obstruction exactness at N (V ) means that [X f V ] N (V ) is mapped to 0 in L n (ZΓ) if and only if it is the image of an element in S(V ) (i.e. can be surgered to an homotopy equivalence).

7 The Browder-Quinn surgery sequence for a smoothly stratified space Let now V be a smoothly stratified pseudomanifold. we bear in mind the Wall s realization of the L-groups we give essentially the same definitions but we require the maps to be stratified and transverse (will come back to definitions) we obtain the Browder-Quinn surgery sequence L BQ n+1 (V ) SBQ (V ) N BQ (V ) L BQ n (V ) There are differences: for example L BQ (V ) depends now on the fundamental groups of all closed strata. Warning: in the paper of Browder and Quinn there are precise statements but no proofs; a few key definitions are also missing. Part of our work was to give a rigorous account.

8 Our program now: explain the Higson-Roe theorem (following Piazza-Schick) say why this is an interesting and useful theorem pass to stratified spaces and explain problems explain how to use analysis on stratified pseudomanifolds in order to achieve the same goal for the Browder-Quinn surgery sequence L BQ n+1 (V ) SBQ (V ) N BQ (V ) L BQ n (V ) assuming V to be a Witt space or more generally a Cheeger space.

9 Higson-Roe analytic surgery sequence change of notation: M is a riemannian manifold with a free and cocompact isometric action of Γ. We write M/Γ for the quotient. Thus, with respect to the previous slides, V = M/Γ and Ṽ = M. we also have a Γ-equivariant complex vector bundle E D c (M) Γ B(L 2 (M, E)) is the algebra of Γ-equivariant bounded operators on L 2 (M, E) that are of finite propagation and pseudolocal D (M) Γ is the norm closure of D c (M) Γ C c (M) Γ B(L 2 (M, E)) is the algebra of Γ-equivariant bounded operators on L 2 (M, E) that are of finite propagation and locally compact C (M) Γ is the norm-closure of C c (M) C (M) Γ is an ideal in D (M) Γ

10 we can consider the short exact sequence (of Higson-Roe); 0 C (M) Γ D (M) Γ D (M) Γ /C (M) Γ 0 and thus K (D (M) Γ ) K (D (M) Γ /C (M) Γ ) δ K +1 (C (M) Γ ) Paschke duality: K (D (M) Γ /C (M) Γ ) K +1 (M/Γ) one can also prove that K (C (M) Γ ) K (C r Γ) these groups behave functorially (covariantly). If ũ : M EΓ is a Γ-equiv. classifying map then we can use ũ to map the Higson-Roe sequence to the universal Higson-Roe sequence: K (C r Γ) K (D Γ ) K +1(BΓ) δ K +1 (C r Γ) where D Γ := D (EΓ) Γ (for simplicity BΓ is a finite complex here). It turns out that δ is the assembly map.

11 Index and rho-classes We assume that we now have a Γ-equivariant Dirac operator D. Let n be the dimension of M. We can define: the fundamental class [D] K n (M/Γ) = K n+1 (D (M) Γ /C (M) Γ ) the index class Ind(D) := δ[d] K n (C (M) Γ ) If D is L 2 -invertible we can use the same definition of [D] but get the rho classes ρ(d) in K n+1 (D (M) Γ ) (no need to go to the quotient) For example if n is odd then ρ(d) = [ 1 2 (1 + D D )] = [Π (D)] K 0 (D (M) Γ )

12 If we only know that Ind(D) = 0 K n (C (M) Γ ) then a perturbation C C (M) Γ such that D + C is L 2 -invertible. can define ρ(d + C) K n+1 (D (M) Γ ) as before; e.g. if n is odd ρ(d + C) := [Π (D + C)] K 0 (D (M) Γ ). notice that ρ(d + C) does depend on C. Atiyah-Patodi-Singer index theory: if W is an oriented manifold with free cocompact action and with boundary W = M then by bordism invariance we know that D has zero index C C ( W ) Γ such that D + C is L 2 -invertible one can prove that there exists an index class Ind(D, C ) K (C (W ) Γ )

13 Mapping surgery to analysis We can now explain the maps Ind, ρ, β in the following diagram L n+1 (ZΓ) S(V ) N (V ) L n (ZΓ) β Ind ρ Ind K n+1 (C (Ṽ )Γ ) K n+1 (D (Ṽ )Γ ) K n (V ) K n (C (Ṽ ) Ind[F : W V [0, 1], r : V BΓ]: use the Hilsum-Skandalis perturbation of F W and take a suitable APS-index class for the signature operator. Well-definedness due to Charlotte Wahl. ρ[x f V ]: use the Hilsum-Skandalis perturbation of f and take the corresponding rho class for the signature operator β[u f V ] := f [ð U sign ] [ðv sign ] Well-definedness of ρ and commutativity of diagram is all in the next Theorem.

14 Theorem (P-Schick) Let C be a trivializing perturbation for D. For the index class Ind(D, C ) K (C (W ) Γ ) the following holds: ι (Ind(D, C )) = j (ρ(d + C )) in K 0 (D (W ) Γ ). Here j : D ( W ) Γ D (W ) Γ is induced by the inclusion W W and ι: C (W ) Γ D (W ) Γ the natural inclusion. Further contributions: P-Schick for Stolz Xie-Yu for Stolz using localization algebras Zenobi (Higson-Roe à la P-Schick for V a topological manifold) Zenobi (Higson-Roe via groupoids) Weinberger-Xie-Yu (Higson-Roe for V a topological manifold) Many beautiful applications (Chang-Weinberger, P-Schick, Weinberger-Yu, Xie-Yu, Zeidler, Zenobi, Weinberger-Xie-Yu...)

15 Stratified pseudomanifolds Above is an example of depth 1;below is an example of depth 2: q p Y 2

16 Basics Let us concentrate on the depth one case. So there is a decomposition of X into two strata: X =Y X Y is the singular set (the bottom blue circle) and X is the regular part (the union of the red cones (without the vertices)). The link of a point p Y is a smooth closed manifold Z (the green circle). A neighborhood of p Y looks like B C(Z), with B a ball in R dim Y.In fact a tubular neighborhood T of Y is a bundle of cones C(Z) T π Y, as in the figure.

17 Examples singular projective algebraic varieties quotients of non-free actions compactifications of locally symmetric spaces moduli spaces

18 Questions and problems: can we run the machine in the singular case? problem 1: for stratified spaces Poincaré duality does not hold consequently, we do not have a signature we do analysis on the regular part X of X ; we need to fix a metric g on X natural metrics are typically incomplete problem 2: the signature operator on Ω c(x ) has many extensions (so, even granting the Fredholm property, which one will be connected to topology?!).

19 Witt spaces We now restrict the class of pseudomanifolds. We consider Witt spaces: Definition X is a Witt space if any even-dimensional link L has dim L/2 IHm (L; Q) = 0. If X is Witt then...everything works!

20 Cheeger spaces We want to drop the Witt assumption and treat more general stratified spaces. We shall treat Cheeger spaces. {Witt spaces} {Cheeger spaces} {Stratified spaces} References: P. Albin, E. Leichtnam, R. Mazzeo, P.P. Hodge theory on Cheeger spaces. Crelle Journal (in press). P. Albin, E. Leichtnam, R. Mazzeo, P.P. The Novikov conjecture on Cheeger spaces. JNCG (in press) P. Albin, M. Banagl, E. Leichtnam, R. Mazzeo, P.P. Refined intersection homology on non-witt spaces. Journal of Analysis and Topology. 2015

21 Iterated conic metrics. Let us concentrate on the depth one case. Recall that a neighborhood of p in the singular set Y (the blue circle) looks like B C(Z), with B a ball in R dim Y. In fact a tubular neighborhood T of Y is a bundle of cones as in figure: C(Z) T π Y If x is the variable along the cone then x = 1 defines a fibration Z H Y A conic metric on X is, by definition, an incomplete metric of the form g := dx 2 + x 2 g Z + π g Y.

22 Closed extensions we want to use Hilbert-space techniques we want closed operators if X is Witt, then d min = d max and ð sign : Ω + c Ω c Ω + c Ω c is essentially self-adjoint in the non-witt case d : Ω k c Ω k+1 c extensions (between d min and d max ) similarly ð sign is NOT essentially self-adjoint has various closed

23 Resolution we resolve the pseudomanifold X to a manifold with corners X (Verona + Brasselet-Hector-Saralegi + ALMP). X has an additional structure: it has an an iterated fibration structure on the boundary (boundary hypersurfaces are fibrations + compatibility relations at the corners between these fibrations). Example: if X is a depth-one space then X is a manifold with boundary and the boundary is our fibration H Y (thus with base equal to the singular stratum (the bottom circle) and fiber the links (the green circles)).

24 Expansions We first consider ð dr := d + d. Recall that a tubular neighborhood T of the singular set Y looks like C(Z) T Y Consider the resolved manifold X ; a manifold with boundary with boundary equal to the fibration Z H Y. If Z is even-dimensional and has cohomology in middle degree then we are NOT in the Witt case. Fundamental Lemma Any u D max (ð dr ) has an asymptotic expansion at Y, u x 1/2 (α 1 (u) + dx β 1 (u)) + ũ with the terms in this expansion distributional: α 1 (u), β 1 (u) H 1/2 (Y ; Λ T Y H f /2 (H/Y )), ũ xh 1 (X, Λ X ) Here H f /2 (H/Y ) is the flat Hodge bundle over Y (with typical fiber H f /2 (Z y )) and f = dim Z

25 Cheeger boundary condition The distributional differential forms α(u), β(u) serve as Cauchy data at Y which we use to define Cheeger ideal boundary conditions. Here is what we do: for any subbundle W H f /2 (H/Y ) Y that is parallel with respect to the flat connection, we define D W (ð dr ) = {u D max (ð dr ) : α 1 (u) H 1/2 (Y ; Λ T Y W ), β 1 (u) H 1/2 (Y ; Λ T Y (W ) )}. We call W a (Hodge) mezzoperversity adapted to g.

26 ANALYTIC RESULTS. Part 1. Every mezzoperversity induces a closed self-adjoint domain D W (ð dr ); (ð dr, D W (ð dr )) is Fredholm with discrete spectrum; We can define a domain for the exterior derivative as an unbounded operator on L 2 differential forms: D W (d); the corresponding de Rham cohomology groups, H W ( X ), are finite dimensional and metric independent; there is a Hodge decomposition theorem.

27 ANALYTIC RESULTS. Part 2 given a mezzoperversity W there is a dual mezzoperversity DW defined in terms of the vertical Hodge- there is a natural non-degenerate pairing HW l ( X ) H n l DW ( X ) R if W = DW then we say that W is self-dual (might not ) X admitting a self-dual mezzoperversity is a Cheeger space; on a Cheeger space we have a non-degenerate pairing HW l ( X ) H n l W ( X ) R and thus a signature σ W ( X ); a self-dual mezzoperversity defines a Fredholm signature operator (ð sign, D W (ð sign )); the index is equal to the signature : σ W ( X ) = ind(ð sign, D W (ð sign )) ind(ð sign,w ) there is a well defined K-homology class [ð sign,w ] in K ( X ) if π 1 ( X ) = Γ and X Γ is the universal cover of X then we also have a higher index class Ind(ð Γ sign,w) K (C ( X Γ ) Γ )

28 Summary+ Crucial Questions Given a Cheeger space X with a fixed self-dual mezzoperversity W and a classifying map r we have defined H W ( X ) σ W ( X ) Z [ð sign,w ] in K ( X ) Ind(ð Γ sign,w ) K (C ( X Γ ) Γ ) = K (C r Γ) Question 1: what happens to these invariants if F : X M is a stratified homotopy equivalence?? Question 2: is there a Hilsum-Skandalis perturbation?? Question 3: can we define the rho class of a stratified homotopy equivalence?? Question 4: how does all this depend on the choice of W??

29 Stratified maps Let F : X M be a smoothly stratified map between depth-1 stratified spaces. We denote by Y X and Y M the singular strata and by T X and T Y the corresponding tubular neighbourhoods. Then F YX : Y X Y M F TX : T X T Y moreover F TX is a bundle map. F is transverse if T X is the pull-back of T Y and F TX is a pull-back map. Pull-back of forms is not L 2 -bounded but we can consider the Hilsum-Skandalis replacement for the pull-back map. We work on the resolved manifold. Proposition Let W be a mezzoperversity for M. Then we can define the pull-back mezzoperversity F (W). If W is self-dual, so is F (W).

30 Stratified homotopy invariance Theorem If F : M M is a stratified homotopy equivalence and W is a mezzoperversity for M then H W( M) H F W ( M ). If W is self-dual σ W ( M) = σ F W( M ) Ind(ð Γ sign,w) = Ind(ð Γ, sign,f W ) K (C r Γ) proved via a Hilsum-Skandalis perturbation.

31 Bordism invariance Theorem Both σ W ( M) and Ind(ð Γ sign,w ) are Cheeger-bordism-invariant: if ( M, W) is bordant to ( M, W ) through (Ẑ, W ) then the Ẑ signature and the index class are the same. Theorem (from an idea of Markus Banagl) Let W and W be two mezzoperversity for M. Then ( M, W) is Cheeger-bordant to ( M, W )

32 Independence on W. The L-class Corollary σ W ( M) and Ind(ð G(r) independent of W!! sign,w ) are stratified homotopy invariant and Consequently: for a Cheeger space M we have a signature and a homology L-class L ( M) H ( M, Q) defined à la Thom. First defined by Banagl using topology. By our Hodge theorem we prove they are equal. Having L ( M) we can define the higher signatures on a Cheeger space {< α, r (L ( M)) >, α H (BΓ, Q)} and formulate the Novikov Conjecture (stratified homotopy invariance) Theorem (Albin-Leichtnam-Mazzeo-P.) If SNC holds for Γ then the Novikov conjecture holds for a Cheeger space M with π 1 ( M) = Γ. In particular it holds for a Witt space.

33 Back to Browder-Quinn I we have seen that on a Cheeger space X with mezzoperversity W there exists a K-homology class [ð sign,w ] K ( X ) if f : M X is a stratified homotopy equivalence then there is an associated Hilsum-Skandalis perturbation for the signature operator on M X with mezzoperversity f W W hence (modulo showing that the operators are in the right algebras) there is a well defined rho class ρ(f, W) K (D ( X Γ ) Γ ) Finally let B be a Cheeger space with boundary and B F X [0, 1] a degree one transverse map. The mezzoperversity W on X, extends trivially to X [0, 1] (call it again W) and pulling it back on B via F we obtain a mezzoperversity F W W on B ( X [0, 1]). If F is a stratified homotopy equivalence then there is a well defined APS-index class Ind APS,W (B F X [0, 1]) K (C ( X Γ ) Γ )

34 Back to Browder-Quinn II Let X be a Cheeger space. Choose a mezzoperversity W. Consider the Browder-Quinn surgery sequence L BQ n+1 ( X ) S BQ ( X ) N BQ ( X ) L BQ Define Ind : L BQ n ( X ) ( X ) on a refined cycle via Ind APS,W Define ρ : S BQ ( X ) K (D ( X Γ ) Γ ) as ρ[ŷ f X ] = ρ(f, W). Define β : N BQ ( X ) K ( X ) as β[ŷ f X ] = f [ð sign,f W] [ð sign,w ]. Theorem These maps are well defined and they are independent of W. Moreover, the following diagram is commutative L BQ n+1 ( X ) S BQ ( X ) N BQ ( X ) L BQ n Ind ρ β K n+1 (C ( X Γ ) Γ ) K n+1 (D ( X Γ ) Γ ) K n ( X ) K n (C

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