Stability of $L^2-$invariants on stratified spaces

TL;DR

This paper extends the understanding of $L^2$-invariants by proving their well-definedness and invariance under certain homotopy equivalences on stratified pseudo-manifolds with wedge metrics.

Contribution

It establishes the well-definedness and invariance of $L^2$-Betti numbers and Novikov-Shubin invariants on stratified pseudo-manifolds, generalizing previous results to Witt pseudo-manifolds.

Findings

$L^2$-Betti numbers are well defined on stratified pseudo-manifolds.

Novikov-Shubin invariants are well defined and invariant under certain homotopies.

Results extend classical invariance theorems to more general stratified spaces.

Abstract

Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_\Gamma$ be a Galois $\Gamma$-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt pseudo-manifolds, we show that the $L^2$-Betti numbers and the Novikov-Shubin invariants are well defined. We then establish their invariance under a smoothly stratified, strongly stratum preserving homotopy equivalence, thus extending results of Dodziuk, Gromov and Shubin to these pseudo-manifolds.