The Novikov conjecture on Cheeger spaces

TL;DR

This paper proves the Novikov conjecture for a class of stratified spaces called Cheeger spaces, using their L2-cohomology and higher signatures, linking analytic and topological invariants.

Contribution

It establishes the Novikov conjecture for Cheeger spaces with fundamental groups satisfying the strong Novikov conjecture, and connects analytic and topological signatures.

Findings

Proves invariance of L2-de Rham cohomology under stratified homotopy.

Shows the analytic signature coincides with the topological signature.

Extends the Novikov conjecture to Cheeger spaces with certain fundamental groups.

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 L2-de Rham cohomology theory satisfying Poincare 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 Mishchenko 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.