Vietoris thickenings and complexes of manifolds are homotopy equivalent

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.