Contractions in persistence and metric graphs

TL;DR

This paper demonstrates that contractions from a space onto a subspace induce embeddings of persistence diagrams, with applications to metric graphs and geodesic spaces, providing a new perspective on circular coordinates.

Contribution

It introduces tight injections of persistence modules and proves that contractions onto loops preserve persistence diagram patterns, extending understanding of topological features in metric spaces.

Findings

Contractions onto shortest loops always exist in metric graphs.

Contractions onto loops induce embeddings of persistence diagrams.

Persistence patterns of $S^1$ appear in larger spaces via contractions.

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 $X$.