Inferring Ambient Cycles of Point Samples on Manifolds with Universal Coverings

TL;DR

This paper presents a method to infer ambient topological features of data sampled from a manifold by leveraging universal coverings, homology, and geodesic matching, enhancing topological data analysis techniques.

Contribution

It introduces a constructive approach using universal coverings and geodesic matching to identify non-trivial loops in the ambient manifold from point cloud data.

Findings

Method successfully identifies non-trivial homology classes

Utilizes universal coverings and geodesic matching

Formalizes approach with groupoids and monodromy

Abstract

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.