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- W4308558739 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 $$kge 0$$ , 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 $$kge 0$$ , where both spaces are defined using open balls (instead of closed balls)." @default.
- W4308558739 created "2022-11-12" @default.
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- W4308558739 date "2022-11-08" @default.
- W4308558739 modified "2023-10-17" @default.
- W4308558739 title "Vietoris thickenings and complexes have isomorphic homotopy groups" @default.
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- W4308558739 doi "https://doi.org/10.1007/s41468-022-00106-5" @default.
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