Contractions in persistence and metric graphs

Abstract

We prove that the existence of a $1$-Lipschitz retraction (a contraction) from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$ embeds into the persistence diagram of $X$. As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture on existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of $S^1$, a contraction $X\to S^1$ implies the same pattern appears in persistence diagram of $X$.

arXiv comment12 pages, 3 figures

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• On Vietoris-Rips complexes of ellipses
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Cited by in corpus (3)

• Critical edges in Rips complexes and persistence
• Contractibility of the Rips complexes of Integer lattices via local domination
• Rigidity of terminal simplices in persistent homology