论文标题
越野河 - 持续的同源性，注入性度量空间和填充半径
Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
论文作者
论文摘要
在应用的代数拓扑界中，越野式式简单过滤引起的持久同源性是一种从度量空间捕获拓扑信息的标准方法。在本文中，我们考虑了一种不同的几何方式来产生持续的度量空间同源性，该方法是首先将给定的度量空间嵌入更大的空间中，然后考虑在此环境度量标准空间内的原始空间的增厚。在此过程中，我们构建了一个适当的类别，用于研究这种持续的同源性概念，并表明，从一个理论意义上讲，越越野里 - 里普斯过滤的标准持续同源性与我们的几何同源性同源性同构是同构的，只要环境指标可以满足所谓的注入性的属性。 作为这种同构结果的应用，我们能够精确地表征出现在任何紧凑型度量空间的越野式核管过滤中的持久性条形码中的间隔类型，也可以简要地证明产品持续的产品和指标空间的度量同源性的表征。我们的结果还允许证明其他公制不变式的越野河流条形码的间隔长度的几个范围。最后，作为另一个应用程序，我们将这种几何持久同源性连接到填充Gromov \ cite \ cite {g07}引入的歧管半径的概念，并显示了与（1）越野越（1）同型越野摩托综合体相关的后果，这些越野摩在越来越多的（2）刻板（2）表征（2）表征（2）表征的结果。来自F.〜Wilhelm的工作。
In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov \cite{G07} and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm.
