论文标题
越野膜增厚和复合物具有同构同型组
Vietoris thickenings and complexes have isomorphic homotopy groups
论文作者
论文摘要
我们研究了与度量空间覆盖物相关的度量增厚与简单复合物之间的关系。令$ \ Mathcal {u} $成为可分开的度量空间$ x $的封面，该套件的均匀直径为单位。越野综合体包含所有简单，其中包含在\ Mathcal {U} $中的一些$ u \中包含的顶点集，而越野尺度增厚是概率度量的空间，并在某些$ u \ in \ Mathcal {U} $中提供支持，配备了最佳的运输量。我们表明，越越野公制的增厚和越野复合物在各个维度上具有同构同型组。特别是，通过适当地选择封面$ \ MATHCAL {U} $，我们可以在越野式公制的同型群体之间获得同构，并在使用``直径$ <r $'''（而不是$ \ le l r r $ r $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $）之间定义了两个空间。同样，我们在聚摄氏度增厚和简单络合物的同型组之间得到同构，其中两个空间都是使用开放球（而不是封闭的球）定义的。
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 contains all simplices with vertex set contained in some $U \in \mathcal{U}$, and the Vietoris metric thickening is the space of probability measures with support in some $U \in \mathcal{U}$, 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 $\mathcal{U}$ 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 $\le r$). 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).
