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

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. 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 [26] 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.

Contractions in persistence and metric graphs

“…The assumptions of our main results of this paper are much easier to verify and in some cases hold more generally. Overall, persistent homology in dimensions 1, 2, and 3 is known to encode some geodesic circles and shortest 1-homology basis by [16,19,21] (and now also by results of this paper), properties of thick-thin decomposition [4] and injectivity radius [15]. On a similar note, the systole of a geodesic space is detected as the first critical scale of persistent fundamental group [16].…”

Approximate and discrete Euclidean vector bundles

“…A possible avenue is to consider the persistent cohomology of the thickened Grassmannians, and use ε-approximate classifying maps to pull back the cohomology classes of these Grassmannians. The question of how long the universal Stiefel-Whintey classes persist in the thickened Grassmannians is related to the filling radius introduced by Gromov ( [19]) and to the generalization considered by Lim, Mémoli, and Okutan in [36]. A related question is whether better bounds for our results can be obtained by using the Vietoris-Rips complex of the Grassmannians, instead of the thickening.…”

Some results about the Tight Span of spheres

“…More precisely, we show that for a tree-like metric space X and for any scale r > 0, each connected component of the neighborhood B r (X, E(X)) is contractible. Then we obtain the result for Vietoris-Rips complex VR 2r (X) by the identification of homotopy type between VR 2r (X) and B r (X, E(X)) in [LMO20].…”

“…In the first place, we became interested in hyperconvex metric spaces because of their application to Topological Data Analysis. In [LMO20], the authors proved that the filtration obtained through increasing thickenings of X inside any hyperconvex metric space is homotopy equivalent to the Vietoris-Rips filtration. Also, the recent papers [JJ19,JJ20] introduced a new notion of curvature for metric spaces through a quantification of their deviation from hyperconvexity, and suggested applications of this notion of curvature for topological/geometric data analysis.…”