Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

Lim, Sunhyuk¹ ; Mémoli, Facundo² ; Okutan, Osman Berat³

¹ Department of Mathematics, Sungkyunkwan University, Suwon-si, South Korea
² Department of Mathematics, The Ohio State University, Columbus, OH, United States
³ Department of Mathematics, Florida State University, Tallahassee, FL, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 2, pp. 1019-1100

Résumé

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. 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, for example the notion of spread introduced by M Katz.

As another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov and show some consequences related to the homotopy type of the Vietoris–Rips complexes of spheres, which follow from work of Katz, and characterization (rigidity) results for spheres in terms of their Vietoris–Rips persistence barcodes, which follow from work of F Wilhelm.

Finally, we establish a sharp version of Hausmann’s theorem for spheres which may be of independent interest.

Détail

DOI : 10.2140/agt.2024.24.1019

Keywords: Vietoris–Rips complexes, persistent homology, filling radius