Footprints of geodesics in 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.

arXiv comment26 pages, 9 figures; Accepted for publication in Mediterranean Journal of Mathematics

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Cited by in corpus (7)

• Contractions in persistence and metric graphs
• The Persistent Topology of Optimal Transport Based Metric Thickenings
• Critical edges in Rips complexes and persistence
• Persistent Homology with Selective Rips complexes detects geodesic circles
• Contractibility of the Rips complexes of Integer lattices via local domination
• Rigidity of terminal simplices in persistent homology
• Geometric Bounds for Persistence