Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius

TL;DR

This paper introduces a geometric approach to persistent homology using ambient space embeddings, establishes an isomorphism with standard Vietoris-Rips homology under injectivity, and applies it to characterize barcode intervals and relate to filling radius.

Contribution

It develops a new geometric framework for persistent homology, proves an isomorphism with classical methods under certain conditions, and applies this to characterize barcode intervals and connect to filling radius.

Findings

Standard persistent homology is isomorphic to geometric persistent homology in injective spaces.

Characterization of intervals in Vietoris-Rips barcodes for compact metric spaces.

Bounds on interval lengths in barcodes using metric invariants.

Abstract

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 inside this ambient metric 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 ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov \cite{G07} and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm.