The Novikov conjecture on Cheeger spaces

Abstract

We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an L 2 -de Rham cohomology theory satisfying Poincaré duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mischenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl. ContentsThe authors are happy to thank Markus Banagl, Jochen Brüning, Jeff Cheeger, Greg Friedman, Michel Hilsum and Richard Melrose for many useful and interesting discussions. Thanks are also due to the referee for a careful reading of the original manuscript and for interesting remarks. 1.Smoothly stratified spaces, iie metrics, and mezzoperversities 1.1. Smoothly stratified spaces. There are many notions of stratified space [Klo09]. Most of them agree that a stratified space X is a locally compact, second countable, Hausdorff topological space endowed with a locally finite (disjoint) decompositionwhose elements, called strata, are locally closed and verify the frontier condition:The depth of a stratum Y is the largest integer k such that there is a chain of strata Y = Y k , . . . , Y 0 with Y j ⊂ Y j−1 for 1 ≤ j ≤ k. A stratum of maximal depth is always a closed manifold. The maximal depth of any stratum in X is called the depth of X as a stratified space. Thus a stratified space of depth 0 is a smooth manifold with no singularity.Where the definitions differ is on how much regularity to impose on the strata and how the strata 'fit together'. In this paper we shall mainly consider smoothly stratified pseudomanifolds with Thom-Mather control data. We proceed to recall the definition, directly taken from [BHS91] and [ALMP12].Definition 1.1. A smoothly stratified space of depth 0 is a closed manifold. Let k ∈ N, assume that the concept of smoothly stratified space of depth ≤ k has been defined. A smoothly stratified space X of depth k + 1 is a locally compact, second countable Hausdorff space which admits a locally finite decomposition into a union of locally closed strata S = {Y j }, where each Y j is a smooth (usually open) manifold, with dimension depending on the index j. We assume the following:

Eta and rho invariants on manifolds with edges

“…There it is proved that the signature operator in the geometric Witt case, twisted by the Mishchenko bundle associated to π : M Γ → M and to the reduced C * -algebra C * r Γ, defines an index class in K * (C * r Γ), a result that should be regarded as a generalisation of the Fredholm property for D itself. The crucial remark in [ALMP12], used again in [ALMP15], is that the microlocal techniques used for D applies equally well for the Mishchenko-twisted version of D once we observe that the Galois covering over a distinguished neighbourhood R b × C(F ) is trivial, see the proof of Proposition 6.3 in [ALMP12]. In this paper we are interested in Von Neumann index theory but the crucial remark applies equally well.…”

“…A precise definition of smoothly stratified spaces of any depth is given in [BHS91], cf. also [ALMP15]. Given a Riemannian metric g B on the edge B, and a symmetric 2-tensor κ on ∂M that restricts to a smooth family of Riemannian metrics on the fibres, the singular edge structure in an neighborhood U of the boundary is given by the Riemannian metric (x denotes the defining function of the boundary)…”

Stratified and unstratified bordism of pseudomanifolds

“…So far, no computations of the bordism groups associated to this class have been carried out. However, there is some evidence to believe that such an inquiry would be profitable: in [4], Banagl studies a class of spaces (now called L-spaces; see [3]) carrying self-dual sheaves compatible with intersection homology. One 15 of the defining properties of these spaces is precisely the vanishing of the signatures of links; these signatures are defined with respect to the sheaf cohomology of the self-dual sheaves (restricted to the link).…”

“…Such invariants could conceivably come from intersection homology groups with perversities that do not meet Goresky and MacPherson's perversity requirements of [17]. One possible such invariant would be the perverse signatures of Hunsicker [23] (see also [15]); another possibility would be the signatures of L-spaces [4,3].…”

“…In Section 5, we study stratified and unstratified bordism groups. It is here that we define IWS classes of pseudomanifolds; these are the classes of pseudomanifolds for which both stratified and unstratified bordism can be defined and for which it is reasonable to compare the two 3 . In particular, these spaces are determined by properties that are intrinsic to the spaces and not their stratifications.…”