Vietoris thickenings and complexes have isomorphic homotopy groups

TL;DR

This paper proves that Vietoris metric thickenings and simplicial complexes have isomorphic homotopy groups, establishing a deep connection between metric and combinatorial topological structures in the context of coverings of metric spaces.

Contribution

It demonstrates the isomorphism of homotopy groups between Vietoris metric thickenings and simplicial complexes, extending classical results to metric measure spaces with optimal transport metrics.

Findings

Vietoris metric thickenings and complexes have isomorphic homotopy groups.

Homotopy groups of Vietoris--Rips thickenings match those of simplicial complexes.

Homotopy groups of Čech metric thickenings are isomorphic to those of simplicial complexes.

Abstract

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris complex contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, equipped with an optimal transport metric. We show that the Vietoris metric thickening and the Vietoris complex have isomorphic homotopy groups in all dimensions. In particular, by choosing the cover $\mathcal{U}$ appropriately, we get isomorphisms between the homotopy groups of Vietoris--Rips metric thickenings and simplicial complexes, where both spaces are defined using the convention ``diameter $< r$'' (instead of $\le r$). Similarly, we get isomorphisms between the homotopy groups of \v{C}ech metric thickenings and simplicial complexes, where both spaces are defined using open balls (instead of closed balls).