Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
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In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the embedding space satisfies a property called hyperconvexity. As an application of this isomorphism we are able to give succint proofs of the characterization of the persistent homology of products and joins of metric spaces. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov <cit.> and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres that follows from work of M. Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence diagrams.
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