Vietoris thickenings and complexes of manifolds are homotopy equivalent
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| Format: | Preprint |
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2025
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| author | Adams, Henry Karassev, Alexandre Virk, Ziga |
| author_facet | Adams, Henry Karassev, Alexandre Virk, Ziga |
| contents | 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 Čech complex to the Čech 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23108 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vietoris thickenings and complexes of manifolds are homotopy equivalent Adams, Henry Karassev, Alexandre Virk, Ziga Geometric Topology General Topology 55N31, 54F45, 54C55, 54C65 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 Čech complex to the Čech 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. |
| title | Vietoris thickenings and complexes of manifolds are homotopy equivalent |
| topic | Geometric Topology General Topology 55N31, 54F45, 54C55, 54C65 |
| url | https://arxiv.org/abs/2512.23108 |