Vietoris thickenings and complexes have isomorphic homotopy groups
Vietoris thickenings and complexes have isomorphic homotopy groups
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DOI：
10.1007/s41468-022-00106-5
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发表时间：
2022-06
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影响因子：
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通讯作者：
Henry Adams;F. Frick;Žiga Virk
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作者：
Henry Adams;F. Frick;Žiga Virk
我们研究了度量加厚与与度量空间覆盖相关的单纯复形之间的关系。设X是可分度量空间X的具有一致直径界的开集的覆盖。Vietoris复形包含所有顶点集在某处的单形，而Vietoris度量加厚是在某处有支集的概率度量空间，具有一个最优传输度量。我们证明了和在所有维度上都有同构同伦群。特别地，通过适当地选择覆盖，我们得到了所有整数的Vietoris-Rips度量加厚的同伦群和单纯复之间的同构，其中这两个空间都是使用约定“直径”(而不是)来定义的。类似地，我们得到了ČECH度量加厚的同伦群和所有整数的单纯复形之间的同构，其中这两个空间都是用开球(而不是闭球)定义的。
We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Letbe a cover of a separable metric spaceXby open sets with a uniform diameter bound. The Vietoris complexcontains all simplices with vertex set contained in some, and the Vietoris metric thickeningis the space of probability measures with support in some, equipped with an optimal transport metric. We show thatandhave isomorphic homotopy groups in all dimensions. In particular, by choosing the coverappropriately, we get isomorphisms between the homotopy groups of Vietoris–Rips metric thickenings and simplicial complexesfor all integers, where both spaces are defined using the convention “diameter” (instead of). Similarly, we get isomorphisms between the homotopy groups of Čech metric thickenings and simplicial complexesfor all integers, where both spaces are defined using open balls (instead of closed balls).