Vietoris thickenings and complexes of manifolds are homotopy equivalent

TL;DR

This paper proves that for finite-dimensional Polish metric spaces, the Vietoris-Rips and Čech complexes are homotopy equivalent to their metric thickenings, with implications for understanding the topology of manifolds.

Contribution

It establishes homotopy equivalences between classical complexes and their metric thickenings for a broad class of spaces, including manifolds.

Findings

Vietoris-Rips complex and metric thickening are homotopy equivalent for finite-dimensional Polish spaces.

Čech complex and its metric thickening are homotopy equivalent under similar conditions.

Metric thickenings of covers of such spaces are strongly locally contractible.

Abstract

We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{VR^m}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence. This occurs, for example, if $X$ is a Riemannian manifold. The same is true for the map $\mathrm{\check{C}}(X;r)$ to $\mathrm{\check{C}}^\mathrm{m}(X;r)$ from the \v{C}ech complex to the \v{C}ech metric thickening, and more generally, for the natural bijection $\mathrm{V}(\mathcal W)\to \mathrm{V^m}(\mathcal W)$ from the Vietoris complex to the Vietoris metric thickening of any uniformly bounded cover $\mathcal W$ of a finite dimensional Polish metric space. We also show that if $X$ is a compact metrizable space, then $\mathrm{V^m}(\mathcal W)$ is strongly locally contractible.