利用万有覆盖推断流形上点样本的环境循环
Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings
AI总结：
本文研究如何利用流形的万有覆盖和点云度量数据，判断单纯复形中的边环是否对应环境流形的非平凡同调类，并在群胚与覆盖单值化框架下给出构造性方法。
摘要：
拓扑数据分析的一个核心目标是从表示为有限点云的数据中识别具有拓扑意义的特征。我们考虑点样本的环境空间为紧黎曼流形的情形。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短测地线相匹配，把复形的一阶同调与周围流形的一阶同调联系起来。在流形的万有覆盖已知的前提下，我们给出一种构造性方法，用于判断复形上给定的边环（或代表一阶同调圈）是否对应于周围流形的非平凡环（或一阶同调类）。我们证明点云上的度量数据及其在覆盖中的纤维足以完成该构造，并在群胚与覆盖的单值化框架中对我们的方法进行了形式化。
英文摘要：
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.
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