Vietoris thickenings and complexes have isomorphic homotopy groups

Henry Adams; Florian Frick; Žiga Virk

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arXiv · 2206.08812

Vietoris thickenings and complexes have isomorphic homotopy groups

Henry Adams·Florian Frick·Žiga Virk

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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).

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Henry Adams, Florian Frick, Žiga Virk. 2022-06-17. Vietoris thickenings and complexes have isomorphic homotopy groups. https://arxiv.org/abs/2206.08812

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