标题： 通过通用覆盖空间推断流形上点样本的环境循环 显示英文标题

标题： Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings

摘要： 拓扑数据分析的核心目标是识别以有限点云形式表示的数据中的拓扑显著特征。 我们考虑点样本的环境空间是一个紧致黎曼流形的情况。 给定在点集上构造的单纯复形，我们可以通过将复形中的边与点之间的最小测地线进行匹配，来将复形的第一同调与环境流形的第一同调联系起来。 如果已知流形的万有覆盖，我们提供一种构造性方法来确定复形上的给定边环（或第一同调循环代表）是否对应于环境流形上的非平凡环（或第一同调类）。 我们证明点云及其在覆盖中的纤维上的度量数据足以进行构造，并在群胚和覆盖的单值性框架中形式化我们的方法。 显示英文摘要

摘要： A central objective of topological data analysis is to identify topologically significant features in data represented as a finite point cloud. We consider the setting where the ambient space of the point sample is a compact Riemannian manifold. Given a simplicial complex constructed on the point set, we can relate the first homology of the complex with that of the ambient manifold by matching edges in the complex with minimising geodesics between points. Provided the universal covering of the manifold is known, we give a constructive method for identifying whether a given edge loop (or representative first homology cycle) on the complex corresponds to a non-trivial loop (or first homology class) of the ambient manifold. We show that metric data on the point cloud and its fibre in the covering suffices for the construction, and formalise our approach in the framework of groupoids and monodromy of coverings.