Footprints of geodesics in persistent homology

TL;DR

This paper explores how subspaces in a metric space generate algebraic elements called footprints in persistent homology, revealing that higher-dimensional homology encodes lower-dimensional geometric features, especially geodesic circles, linking persistent homology with Riemannian geometry.

Contribution

It introduces the concept of footprints generated by subspaces in persistent homology and shows how geodesic circles can produce detectable features in higher-dimensional homology.

Findings

Footprints typically appear in dimensions above that of the subspace.

Geodesic circles can generate non-trivial footprints in odd and two-dimensional homology.

Some contractible geodesics can be detected using 2- and 3-dimensional persistent homology.

Abstract

Given a metric space $X$ and a subspace $A\subset X$, we prove $A$ can generate various algebraic elements in persistent homology of $X$. We call such elements (algebraic) footprints of $A$. Our results imply that footprints typically appear in dimensions above the dimension of $A$. Higher-dimensional persistent homology thus encodes lower-dimensional geometric features of $X$. We pay special attention to a specific type of geodesics in a geodesic surface $X$ called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and length spectrum in Riemannian geometry.