Vietoris thickenings and complexes have isomorphic homotopy groups

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 \({\mathcal {V}}({\mathcal {U}})\) contains all simplices with vertex set contained in some \(U \in {\mathcal {U}}\), and the Vietoris metric thickening \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) is the space of probability measures with support in some \(U \in {\mathcal {U}}\), equipped with an optimal transport metric. We show that \({\mathcal {V}}^\textrm{m}({\mathcal {U}})\) and \({\mathcal {V}}({\mathcal {U}})\) 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 \(\pi _k(\textrm{VR}^\textrm{m}(X;r))\cong \pi _k(\textrm{VR}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using the convention “diameter \(< r\)” (instead of \(\le r\)). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexes \(\pi _k(\check{\mathrm {{C}}}^\textrm{m}(X;r))\cong \pi _k(\check{\mathrm {{C}}}(X;r))\) for all integers \(k\ge 0\), where both spaces are defined using open balls (instead of closed balls).