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

TL;DR

This work develops a geometric counterpart to Vietoris-Rips persistence by embedding a metric space into an injective ambient space and analyzing thickenings inside that ambient space. The authors prove an Isomorphism Theorem showing the Vietoris-Rips filtration is categorically equivalent to a geometric filtration built from ambient thickenings, with the ambient space $L^ olinebreak[4]{}^ olinebreak[4]{}{ olinebreak[4]}\infty(X)$ as a canonical example; more generally, any injective space suffices. This framework yields practical results: it provides a complete interval form for Vietoris-Rips barcodes, extends products and gluings via a persistent Künneth-type formula, and enables precise links between persistence and the filling radius, hyperbolicity, and rigidity of spheres. By connecting TDA invariants to intrinsic metric invariants, the approach bridges applied topology and quantitative geometry, producing both theoretical characterizations and tools for lower bounds on Gromov–Hausdorff distances. The paper further analyzes homotopy types of VR_r(X) for spheres and complex projective spaces, refinements for the geodesic and $\ell^\infty$-metrics, and robust stability results for the geometric filtration under metric distortion. Altogether, the framework broadens the scope of Vietoris-Rips analysis, enabling sharper geometric bounds and rigidity phenomena for a wide class of spaces.

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.

Figures (12)

• Figure 1: $B_r(\mathbb{S}^1_\infty,\mathbb{D}^2_\infty)=\mathbb{D}^2_\infty\backslash V_{2,r}$ in the plane $\mathbb{R}^2_\infty$. The set $V_{2,r}$ is given by the intersection of the 4 closed disks shown in the figure. See Theorem \ref{['thm:ncircletightspan']}.
• Figure 2: The construction of the one parameter family of surfaces $\mathbb{S}^2_h$ with the same filling radius as $\mathbb{S}^2$. $u_1$, $u_2$, $u_3$ and $u_4$ are vertices of a regular geodesic tetrahedron, $T$ is a small geodesic triangle, which is used to form a cylinder of height $h$ (left figure). See Example \ref{['ex:one-parameter']} for details.
• Figure 3: A 2-dimensional sphere with more than one interval in its $\mathrm{barc}^\mathrm{VR}_2$.
• Figure 4: The points $p,p_0$ and $p_1$ in the proof of Lemma \ref{['lemma:nsquaretightspan']}.
• Figure 5: In this illustration, $s<s'$ and $u_{(\omega,s)}>u_{(\omega',s')}.$ See Remark \ref{['rem:spec-uv']}.

Theorems & Definitions (263)

• Theorem 1: Isomorphism Theorem
• Proposition 1.1
• Theorem 1
• Theorem 1
• Corollary 1.1: $\dgm^\vr_*$ rigidity for spheres
• Definition 1: Vietoris-Rips filtration
• Definition 2: Persistence family
• Definition 3: Persistence module
• Definition 4: Interval persistence module, carlsson2010zigzag
• Definition 5: Barcode