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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.

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
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