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

Sunhyuk Lim, Facundo Mémoli and Osman Berat Okutan

Algebraic & Geometric Topology 24 (2024) 1019–1100
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. 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.

Keywords
Vietoris–Rips complexes, persistent homology, filling radius
Mathematical Subject Classification
Primary: 53C23, 55N31
References
Publication
Received: 18 January 2022
Revised: 16 July 2022
Accepted: 10 August 2022
Published: 12 April 2024
Authors
Sunhyuk Lim
Department of Mathematics
Sungkyunkwan University
Suwon-si
South Korea
https://sites.google.com/view/sunhyuklim
Facundo Mémoli
Department of Mathematics
The Ohio State University
Columbus, OH
United States
http://facundo-memoli.org
Osman Berat Okutan
Department of Mathematics
Florida State University
Tallahassee, FL
United States
https://sites.google.com/view/ookutan/home

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