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Footprints of Geodesics in Persistent Homology

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Abstract

Given a metric space X and a subspace \(A\subset X\), we prove that 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 \(\dim (A)\). 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 the length spectrum in Riemannian geometry.

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  1. University of Ljubljana, Ljubljana, Slovenia

    Žiga Virk

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  1. Žiga Virk
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Correspondence to Žiga Virk.

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Research was supported by Slovenian Research Agency under Grant No. N1-0114 and P1-0292. The author would like to thank the referee for a careful reading and useful comments.

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Virk, Ž. Footprints of Geodesics in Persistent Homology. Mediterr. J. Math. 19, 160 (2022). https://doi.org/10.1007/s00009-022-02089-0

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  • DOI: https://doi.org/10.1007/s00009-022-02089-0

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