The Novikov conjecture on Cheeger spaces. (English) Zbl 1375.57034
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. It is shown that this cohomology theory is invariant under stratified homotopy equivalence and that its signature is invariant under Cheeger space cobordism. Using coupling with Mishchenko bundles, the authors define higher analytic signatures for a Cheeger space and prove their stratified homotopy invariance under the additional assumption that the assembly map is rationally injective. Then they show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl. Thus, the Novikov conjecture is proved for oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture.
Reviewer: Vladimir M. Manuilov (Moskva)
MSC:
57R19 | Algebraic topology on manifolds and differential topology |
58J20 | Index theory and related fixed-point theorems on manifolds |
46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |
46L85 | Noncommutative topology |
19K56 | Index theory |
32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
58J40 | Pseudodifferential and Fourier integral operators on manifolds |
58B34 | Noncommutative geometry (à la Connes) |
58A35 | Stratified sets |
57P99 | Generalized manifolds |