Contractions in Persistence and Metric Graphs

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