Vietoris-Rips persistent homology, injective metric spaces, and the filling radius (Q6536732)

The title of the article refers to an article by \textit{L. Vietoris} [Math. Ann. 97, 454--472 (1927; JFM 53.0552.01)] and to another article by \textit{M. Gromov} [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75--263 (1987; Zbl 0634.20015)]. Eliyahu Rips' name is also mentioned but no article of his is specified. But the notion of Vietoris-Rips complex appeared previously in an article by \textit{J.-C. Hausmann} [Ann. Math. Stud. 138, 175--188 (1995; Zbl 0928.55003)]. And for the notion of persistent homology the authors cite a series of articles, by several authors, from the period 1990--2000. \N\NFor a metric space \((X,d_X)\) and \(r>0\), the \(r\)-Vietoris-Rips complex \(VR_r(X)\) has \(X\) as its vertex set, and simplexes are all nonempty finite subsets of \(X\) whose diameter is strictly less than \(r\). This is important for a Riemannian manifold \(M\) because there is a constant \(r(M)\) so that \(VR_r(M)\) is homotopy equivalent to \(M\) for any \(r\in (0,r(M))\). This allows references to the homology and cohomology of the manifold \(M\). \N\NIf \(r\leq s\), then \(VR_r(X)\subset VR_s(X)\), so that \(VR_\ast(X)\) is a filtration called the open Vietoris-Rips filtration of \(X\). The notion of Vietoris-Rips persistent homology is a particular case of the following definition: A persistence family is a collection \((U_r,f_{r,s})_{r\leq s\in T}\), where \(T\) is a nonempty subset of \(\mathbb{R}\) such that, for each \(r\leq s\leq t\in T\), \(U_r\) is a topological space and \(f_{r,s}:U_r\rightarrow U_s\) a continuous map, such that \(f_{r,r}=id_{U_r}\), \(f_{s,t}\circ f_{r,s}=f_{r,t}\). The morphism between two persistence families indexed by the same \(T\subseteq \mathbb{R}\) is defined naturally. Also naturally the notion of persistence module is defined as a functor from the category \((T,\leq)\) of sets to the category of modules. Relative to homology, for \(k\geq 0\), applying the homology functor \(H_k(-,\mathbb{F})\) to a persistence family \((U_r,f_{r,s})_{r\leq s\in T}\) defines a persistence module \(H_k(U_\ast;\mathbb{F})\), and this correspondence is functorial. This type of homology is called persistent homology. Finally, the homology functor is applied to the Vietoris-Rips filtration of a metric space \(X\) and this induces a persistence module (with \(T=\mathbb{R}_{>0}\)) where the morphisms are those induced by inclusions. This persistence module is denoted by \(PH_k(VR_\ast(X),\mathbb{F})\) and this is called the \textsl{Vietoris-Rips persistent homology} of \(X\). \N\NThe present article is very voluminous in terms of the number of pages, but it also has many new and interesting original results. In addition, this extensive article can serve as a manual for those who want to learn about this interesting subject, finding in it the notions and theoretical results and their applications and usefulness. \N\NRegarding the new contributions of the authors in this article: ``One main contribution of this paper is establishing a precise relationship (i.e. a filtred homotopy equivalence) between the Vietoris-Rips simplicial filtration of a metric space and a more geometric (or extrinsic) way of assigning a persistence module to a metric space, which consists of first isometrically embedding it into a larger space and then considering the persistent homology of the filtration obtained by considering the resulting system of nested neighborhoods of the original space inside this ambient space. These neighborhoods, being also metric (and thus topological) spaces, permit giving a short proof for the Künneth formula for Vietoris-Rips persistent homology''. A nice example is the Kuratowski isometric embedding of a compact metric space \((X,d_X)\) in \(L^\infty(X)\), the Banach space consisting of all the bounded real-valued functions on \(X\), together with the \(\ell^\infty\)-norm. \N\NWith the introduction, the article has nine sections. In Section 2, some necessary definitions and results about Vietoris-Rips filtration, persistence, and injective metric space are given. In Section 3, a category of metric pairs that is necessary for an extrinsic persistent homology is obtained. In Section 4 it is proved that the Vietoris-Rips filtration can be seen as a special case of persistent homology obtained through metric embeddings (Theorem 4.1). Sections 5--9 provide applications of the isomorphism theorem 4.1. For example, in Section 6 a new proof of a formula about the Vietoris-Rips persistence of metric products and metric gluing of metric spaces is given. In Section 7 several results concerning the homotopy types of Vietoris-Rips filtrations of spheres and complex projective spaces are proved. In Section 8 the authors give a new proof of Rips and Gromov's result about the contractibility of the Vietoris-Rips complex of hyperbolic geodesic metric spaces. In Section 9, ``we give some applications of our ideas to the filling radius of Riemannian manifolds and also study consequences related to the characterization of spheres by their persistence barcodes and some generalizations and novel stability properties of the filling radius''. The Appendix contains proofs and some background material.