2022/06/17 by Henry Adams, Florian Frick, Adams, Henry +3 · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2206.08812
openalex publication_date 2022/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let 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 ∈ U, and the Vietoris metric thickening is the space of probability measures with support in some U ∈ 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 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 ≤ r). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes, where both spaces are defined using open balls (instead of closed balls).