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

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.


Introduction
The simplicial complex nowadays referred to as the Vietoris-Rips complex was originally introduced by Leopold Vietoris in the early 1900s in order to build a homology theory for metric spaces [72]. Later, Eliyahu Rips and Mikhail Gromov [43] both utilized the Vietoris-Rips complex in their study of hyperbolic groups.
Given a metric space (X, d X ) and r > 0 the r-Vietoris-Rips complex VR r (X) has X as its vertex set, and simplices are all those nonempty finite subsets of X whose diameter is strictly less than r. In [46], Hausmann showed that the Vietoris-Rips complex can be used to recover the homotopy type of a Riemannian manifold M . More precisely, he introduced a quantity r(M ) (a certain variant of the injectivity radius), and proved that VR r (M ) is homotopy equivalent to M for any r ∈ (0, r(M )).
Since VR r (X) ⊆ VR s (X) for all 0 < r ≤ s, this construction then naturally induces the so called Vietoris-Rips simplicial filtration of X, denoted by VR * (X) = VR r (X) r>0 . By applying the simplicial homology functor (with coefficients in a given field) one obtains a persistence module: a directed system V * = V r vrs − V s r≤s of vector spaces and linear maps (induced by the simplicial inclusions). The persistent module obtained from VR * (X) is referred to as the Vietoris-Rips persistent homology of X.
The notion of persistent homology arose from work by Frosini, Ferri, and Landi [36,37], Robins [68], and Edelsbrunner [23,33] and collaborators. After that, considering the persistent homology of the simplicial filtration induced from Vietoris-Rips complexes was a natural next step. For example, Carlsson and de Silva [15] applied Vietoris-Rips persistent homology to topological estimation from point cloud data, and Ghrist and de Silva applied it to sensor networks [24]. Its efficient computation has been addressed by Bauer in [9]. A more detailed historical survey and review of general ideas related to persistent homology can be found in [14,31,32].
The persistent homology of the Vietoris-Rips filtration of a metric space provides a functorial way 1 of assigning a persistence module to a metric space. Persistence modules are usually represented, up to isomorphism, as barcodes: multisets of intervals each representing the lifetime of a homological feature. In this paper, the barcodes are associated to Vietoris-Rips filtrations, and these barcodes will be denoted by barc VR * (·). In the areas of topological data analysis (TDA) and computational topology, this type of persistent homology is a widely used tool for capturing topological properties of a dataset [9,15,24].
Despite its widespread use in applications, little is known in terms of relationships between Vietoris-Rips barcodes and other metric invariants. For instance, whereas it is obvious that the right endpoint of any interval I in barc VR * (X) must be bounded above by the diameter of X, there has been little progress in relating the length of bars to other invariants such as volume (or Hausdorff measure) or curvature (whenever defined).
Contributions. One main contribution of this paper is establishing a precise relationship (i.e. a filtered 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 1 Where for metric spaces X and Y morphisms are given by 1-Lipschitz maps φ : X Y , and for persistence modules V * and W * morphisms are systems of linear maps ν * = (ν r : V r W r ) r>0 making all squares commute.
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 of the Künneth formula for Vietoris-Rips persistent homology.
A particularly nice ambient space inside which one can isometrically embed any given compact metric space (X, d X ) is L ∞ (X); the Banach space consists of all the bounded real-valued functions on X, together with the ∞ -norm. The embedding is given by X x d X (x, ·): it is indeed immediate that this embedding is isometric since d X (x, ·) − d X (x , ·) ∞ = d X (x, x ) for all x, x ∈ X. This is usually called the Kuratowski isometric embedding of X.
That the Vietoris-Rips filtration of a finite metric space produces persistence modules isomorphic to the sublevel set filtration of the distance function was already used in [18] in order to establish the Gromov-Hausdorff stability of Vietoris-Rips persistence of finite metric spaces.
In this paper we significantly generalize this point of view by proving an isomorphism theorem between the Vietoris-Rips filtration of any compact metric space X and its Kuratowski filtration: r>0 , a fact which immediately implies that their persistent homologies are isomorphic.
We do so by constructing a filtered homotopy equivalence between the Vietoris-Rips filtration and the sublevel set filtration induced by δ X . Furthermore, we prove that L ∞ (X) above can be replaced with any injective (or equivalently, hyperconvex) metric space [28,56] admitting an isometric embedding of X: Theorem 5 (Isomorphism Theorem). Let η : Met PMet be a metric homotopy pairing (for example the Kuratowski functor). Then B * • η : Met hTop * is naturally isomorphic to VR 2 * .
Above, Met is the category of compact metric spaces with 1-Lipschitz maps, PMet is the category of metric pairs (X, E) where X E isometrically, E is an injective metric space, a metric homotopy pairing is any right adjoint to the forgetful functor (e.g. the Kuratowski embedding), and B * is the functor sending a pair (X, E) to the filtration B r (X, E) r>0 ; see Sections 3 and 4.
A certain well known construction which involves the isometric embedding X L ∞ (X) is that of the filling radius of a Riemannian manifold [42] defined by Gromov in the early 1980s. In that construction, given an n-dimensional Riemannian manifold M one studies for each r > 0 the inclusion ι r : M δ −1 M ([0, r)), and seeks the infimal r > 0 such that the map induced by ι r at n-th homology level annihilates the fundamental class [M ] of M . This infimal value defines FillRad(M ), the filling radius of M .
Via our isomorphism theorem we are able prove that there always exists a bar in the barcode of a manifold whose length is exactly twice its filling radius: As a step in his proof of the celebrated systolic inequality, Gromov proved in [42] that the filling radius satisfies FillRad(M ) ≤ c n vol(M ) 1/n for any M dimensional complete manifold (where c n is a universal constant, and Nabutovsky recently proved that c n can be improved to n 2 [64,Theorem 1.2]). This immediately yields a relationship between barc VR * (M ) and the volume of M . The fact that the filling radius has already been connected to a number of other metric invariants also permits importing these results to the setting of Vietoris-Rips barcodes (see Section 9.3). This in turn permits relating barc VR * (M ) with other metric invariants of M , a research thread which has remained mostly unexplored. See Proposition 9.7 for a certain generalization of Proposition 9.4 to ANR spaces.
In a series of papers [50,51,52,53] M. Katz studied both the problem of computing the filling radius of spheres (endowed with the geodesic distance) and complex projective spaces, and the problem of understanding the change in homotopy type of δ −1 X ([0, r)) when X ∈ {S 1 , S 2 } as r increases.
Of central interest in topological data analysis has been the question of providing a complete characterization of the Vietoris-Rips persistence barcodes of spheres of different dimensions. Despite the existence of a complete answer to the question for the case of S 1 [3] due to Adams and Adamaszek, relatively little is known for higher dimensional spheres. In [4] the authors consider a variant of the Vietoris-Rips filtration, which they call Vietoris-Rips metric thickening. The authors are able to obtain information about the succesive homotopy types of this filtration on spheres of different dimension (see Section 5 of [4]) for a certain range of values of the scale parameter.
The authors of [4] conjecture that the open Vietoris-Rips filtration (which is the one considered in the present paper) is filtered homotopy equivalent to their open Vietoris-Rips metric thickening filtration (as a consequence their persistent homologies are isomorphic); weak forms of this conjecture are contained in Remark 3.3, Conjecture 6.12 of [4], and Corollary 5.8 of [2].
Our isomorphism theorem (Theorem 5) permits applying Katz's results in order to provide partial answers to the questions mentioned above and also to elucidate other properties of the standard open Vietoris-Rips filtration and its associated persistence barcodes barc VR * (·). In addition to these results derived from our isomorphism theorem, in Appendix A.4, we refine certain key lemmas used in the original proof of Hausmann's theorem [46] and establish the homotopy equivalence between VR r (S n ) and S n for any r ∈ 0, arccos − 1 n+1 : Theorem 10. For any n ∈ Z >0 , we have VR r (S n ) S n for any r ∈ 0, arccos −1 n+1 . Note that this is indeed an improvement since, for spheres, Hausmann's quantity satisfies r(S n ) = π 2 < arccos − 1 n+1 . This improvement is obtained with the aid of a refined version of Jung's theorem (cf. Theorem 13) which we also establish. Theorem 10 also improves upon [50,Remark p.508], see discussion in Section 7.1.
In the direction of characterizing the Vietoris-Rips barcodes of spheres, we are able to provide a complete characterization of the homotopy types of the Vietoris-Rips complexes of round spheres S n−1 ⊂ R n endowed with the (restriction of the) ∞ -metric, which we denote by S n−1 ∞ . Two critical observations are that (1) the r-thickening of S n−1 ∞ inside of R n ∞ (R n equipped with the ∞ -metric) is homotopy equivalent to the r-thickening of S n−1 ∞ inside of D n ∞ (n-dimensional unit ball with ∞ -metric), and (2) that it is easier to find the precise shape of the latter: where V n,r := (p 1 ,...,pn)∈{r,−r} n (x 1 , . . . , x n ) ∈ R n : In particular, for r > 1 √ n we have V n,r = ∅ so that B r (S n−1 ∞ , D n ∞ ) = D n ∞ . As a result, B r (S n−1 ∞ , R n ∞ ) is homotopy equivalent to S n−1 for r ∈ 0, 1 √ n and contractible for r > 1 √ n . (See Figure 1 for an illustration for the case when n = 2) From a different perspective, by appealing to our isomorphism theorem, it is also possible to apply certain results from quantitative topology to the problem of characterization of metric spaces by their Vietoris-Rips persistence barcodes. In applied algebraic topology, a general question of interest is: Question 1. Assume X and Y are compact metric spaces such that barc VR k (X; F) = barc VR k (Y ; F) for all k ∈ Z ≥0 . Then, how similar are X and Y (in a suitable sense)?
It follows from work by Wilhelm [75] and Yokota [76] on rigidity properties of spheres via the filling radius, and the isomoprhism theorem (Theorem 5), that any n-dimensional Alexandrov space without boundary and sectional curvature bounded below by 1 such that its Vietoris-Rips persistence barcode agrees with that of S n must be isometric to S n . This provides some new information about the inverse problem for persistent homology [22,39]. More precisely, and for example, we obtain the corollary below, where for an n-dimensional manifold M , I M n denotes the persistence interval in barc VR n (M ) induced by the fundamental class of M (cf. Proposition 9.4): Corollary 9.5 (barc VR * rigidity for spheres). For any closed connected n-dimensional Riemannian manifold M with sectional curvature K M ≥ 1, we have: 2. If I M n = I S n n then M is isometric to S n .
3. There exists n > 0 such that if length(I S n n ) − n < length(I M n ), then M is diffeomorphic to S n . 4. If length(I M n ) > π 3 , then M is a twisted n-sphere (and in particular, homotopy equivalent to the n-sphere).
The lower bound on sectional curvature is crucial -in Example 9.6 we construct a one parameter family of deformations of the sphere S 2 with constant filling radius (cf. Figure 2). See Propositions 9.8 and 9.9 for additional related results.
The authors hope that this paper can help bridge between the Applied Algebraic Topology and the Quantitative Topology communities.
Organization. In Section 2, we provide some necessary definitions and results about Vietoris-Rips filtration, persistence, and injective metric spaces.
In Section 3, we construct a category of metric pairs. This category will be the natural setting for our extrinsic persistent homology. Although being functorial is trivial in the case of Vietoris-Rips persistence, the type of functoriality which one is supposed to expect in the case of metric embeddings is a priori not obvious. We address this question in Section 3 by introducing a suitable category structure.
In Section 4, we show that the Vietoris-Rips filtration can be (categorically) seen as a special case of persistent homology obtained through metric embeddings via the isomorphism theorem (Theorem 5). In this section, we also we also establish the stability of the filtration obtained via metric embeddings.
In Section 5, we prove that any interval in persistence barcode for open Vietoris-Rips filtration must have open left endpoint and closed right endpoint.
In Section 6, we obtain new proofs of formulas about the Vietoris-Rips persistence of metric products and metric gluings of metric spaces.
In Section 7, we prove a number of results concerning the homotopy types of Vietoris-Rips filtrations of spheres and complex projective spaces. Also, we fully compute the homotopy types of Vietoris-Rips filtration of spheres with ∞ -norm.
In Section 8, we reprove Rips and Gromov's result about the contractibility of the Vietoris-Rips complex of hyperbolic geodesic metric spaces, by using our method consisting of isometric embeddings into injective metric spaces. As a result, we will be able to bound the length of intervals in Vietoris-Rips persistence barcode by the hyperbolicity of the underlying space.
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 relegated proofs and some background material.

Background
In this section we cover the background needed for proving our main results. We alert readers that, in this paper, the same notation can mean either a simplicial complex itself or its geometric realization, interchangeably. The precise meaning will be made clear in each context.

Vietoris-Rips filtration and persistence
References for the definitions and results in this subsection are [10,57].
Definition 1 (Vietoris-Rips filtration). Let X be a metric space and r > 0. The (open) Vietoris-Rips complex VR r (X) of X is the simplicial complex whose vertices are the points of X and whose simplices are those finite subsets of X with diameter strictly less then r. Note that if r ≤ s, then VR r (X) is contained in VR s (X). Hence, the family VR * (X) is a filtration, called the open Vietoris-Rips filtration of X.
Definition 2 (Persistence module). A persistence module V = (V r , v r,s ) 0<r≤s over R >0 is a family of F-vector spaces V r>0 for some field F with morphisms v r,s : V r V s for each r ≤ s such that In other words, a persistence module is a functor from the poset (R >0 , ≤) to the category of vector spaces. The morphisms v * , * are referred to as the structure maps of V * .
By 0 * we will denote the zero persistence module.
Definition 3 (Persistence families). A persistence family is a collection U r , f r,s 0<r≤s such that for each 0 < r ≤ s ≤ t, U r is a topological space, f r,s : U r U s is a continuous map, f r,r = id Ur and f s,t • f r,s = f r,t .
Given two persistence families (U * , f * , * ) and (V * , g * , * ) a morphism from the first one to the second is a collection (φ * ) r>0 such that for each 0 < r ≤ s, φ r is a homotopy class of maps from U r V r and φ s • f r,s is homotopy equivalent to g r,s • φ r .
Remark 2.1 (Persistent Homology). For any k ≥ 0, applying the k-dimensional homology functor (with coefficients in a field F) to a persistence family U r , f r,s 0<r≤s produces the persistence module H k (U * ; F) where the morphisms are induced by f r,s 0<r≤s . Following the extant literature, we will use the term persistent homology of a persistent family (a.k.a. a filtration) to refer to the persistence module obtained upon applying the homology functor to this family.
In particular, one can apply the homology functor to a Vietoris-Rips filtration of a metric space X. This induces a persistence module where the morphisms are induced by inclusions. As a persistence module, it is denoted by PH k (VR * (X); F).
Definition 4 (Indecomposable persistence modules, [16]). Given an interval I in R >0 and a field F, the persistence module I F is defined as follows: The vector space at r is F if r is in I and zero otherwise. Given r ≤ s, the morphism corresponding to (r, s) is the identity if r, s are in I and zero otherwise.
We need a mild regularity condition to get well-defined notion of barcode, see [20].
Definition 5 (q-tame persistence module). A persistence module V r , v r,s 0<r≤s is said to be qtame if rank(v r,s ) < ∞ whenever r < s. Remark 2.2. In [19,Proposition 5.1], the authors proved that if X is a totally bounded metric space, then PH k (VR * (X); F) is q-tame for any nonnegative integer k ≥ 0 and any field F. Theorem 1 ( [20]). If V * is a q-tame persistence module, then there is a family of intervals (I λ ) λ∈Λ , unique up to reordering, such that V * is isomorphic to λ (I λ ) F . Definition 6 (Barcode). A barcode is a multiset of intervals in R. By Theorem 1, there exists a barcode associated to each q-tame persistence module V * which we denote by barc(V * ). This is called the persistence barcode associated to the persistence module V * . If X is a totally bounded metric, then we denote the barcode corresponding to PH k (VR * (X); F) by barc VR k (X; F).
Definition 7 (Interleaving distance). Two persistence modules V * and W * are said to be δ-interleaved for some δ ≥ 0 if there are natural transformations f : V * W * +δ and g : W * V * +δ such that f • g and g • f are equal to the structure maps W * W * +2δ and V * V * +2δ , respectively. The interleaving distance between V * and W * is defined as It is known [10] that d I is an extended pseudo-metric on the collection of all persistence modules.
Example 2.1. Consider 0 * , the 0 persistence module. Then, for any finite dimensional V * one has Definition 8 (Bottleneck distance). Let M and M be two barcodes. A subset P ⊆ M × M is said to be a partial matching between M and M if it satisfies the following constraints: • every interval I ∈ M is matched with at most one interval of M , i.e. there is at most one interval I ∈ M such that (I, I ) ∈ P , • every interval I ∈ M is matched with at most one interval of M , i.e. there is at most one interval I ∈ M such that (I, I ) ∈ P .
The bottleneck distance between M and M is defined as Theorem 2 (Isometry theorem, [20,Theorem 5.14]). For any two q-tame persistence modules V * and W * , the following equality holds: For the proof of the following theorem, see [19,Lemma 4.3] or [11,18,60].

Injective (Hyperconvex) metric spaces
A hyperconvex metric space is one where any collection of balls with non-empty pairwise intersections forces the non-empty intersection of all balls. These were studied by Aronszajn and Panitchpakdi [7] who showed that every hyperconvex space is an absolute 1-Lipschitz retract. Isbell [48] proved that every metric space admits a smallest hyperconvex hull (cf. the definition of tight span below). Dress rediscovered this concept in [27] and subsequent work provided much development in the context of phylogenetics [70,28]. More recently, in [49] Joharinad and Jost considered relaxations of hyperconvexity and related it to a certain notion of curvature applicable to general metric spaces.
Definition 9 (Injective metric space). A metric space E is called injective if for each 1-Lipschitz map f : X E and isometric embedding of X intoX, there exists a 1-Lipschitz mapf :X E extending f : The following lemma is easy to deduce from the definition of hyperconvex space. Lemma 2.1. Any nonempty intersection of closed balls in hyperconvex space is hyperconvex.
For a proof of the following proposition, see [7]  Moreover, every injective metric space is a contractible geodesic metric space, as one can see in Lemma 2.2 and Corollary 2.1.
Proof. By Lemma 2.2, there is a geodesic bicombing γ on E. Fix arbitrary point x 0 ∈ E. Then, restricting γ to E × {x 0 } × [0, 1] gives a deformation retraction of E onto x 0 . Hence E is contractible.
Example 2.2. For any set S, the Banach space L ∞ (S) consisting of all the bounded real-valued functions on S with the ∞ -norm is injective.
is an isometric embedding and it is called the Kuratowski embedding. Hence every compact metric space can be isometrically embedded into an injective metric space.
Let us introduce some notations which will be used throughout this paper. Suppose that X is a subspace of a metric space (E, d E ). For any r > 0, let B r (X, As one more convention, whenever there is an isometric embedding ι : X − E, we will use the notation B r (X, E) instead of B r (ι(X), E). For instance, in the sequel we will use B r (X, L ∞ (X)) rather than B r (κ(X), L ∞ (X)).
Definition 13. For any metric space E, a nonempty subspace X, and r > 0, theČech complex C r (X, E) is defined as the nerve of the open covering U r := {B r (x, E) : x ∈ X}. In other words, C r (X, E) is the simplicial complex whose vertices are the points of X, and {x 0 , . . . , x n } ⊆ X is a simplex inČ r (X, E) if and only if ∩ n i=0 B r (x i , E) = ∅.
The following observation is simple, yet it plays an important role in our paper.
is an injective metric space, and ∅ = X ⊆ E then, for any r > 0, C r (X, E) = VR 2r (X). Remark 2.3. Note that Proposition 2.2 is optimal in the sense that ifČ r (X, E) = VR 2r (X) holds true for all ∅ = X ⊆ E, then this condition itself resembles hyperconvexity of E (cf. Definition 10). Also note that Proposition 2.2 is a generalization of both [41,Lemma 4] and [18,Lemma 2.9] in that those papers only consider the case when X is finite and E = ∞ (X).
Proof of Proposition 2.2. Because of the triangle inequality, it is obvious thatČ r (X, E) is a subcomplex of VR 2r (X). Now, fix an arbitrary simplex {x 0 , . . . , x n } ∈ VR 2r (X). It means that d E (x i , x j ) < 2r for any i, j = 0, . . . , n. Now, since E is hyperconvex by Proposition 2.1, there exists x ∈ E such that d X (x i , x) < r for any i = 0, . . . , n (note that, since {x 0 , . . . , x n } is finite, one can use < instead of ≤ when invoking the hyperconvexity property). Therefore, {x 0 , . . . , x n } ∈Č r (X, E). Hence VR 2r (X) is a subcomplex ofČ r (X, E). This concludes the proof.
In particular, Proposition 2.2 implies the following result. Proposition 2.3. Let X be a subspace of an injective metric space (E, d E ). Then, for any r > 0, the Vietoris-Rips complex VR 2r (X) is homotopy equivalent to B r (X, E).
The proof of Proposition 2.3 will use the following lemma: In an injective metric space E, every non-empty intersection of open balls is contractible.
Proof. Let γ be a geodesic bicombing on E, whose existence is guaranteed by Lemma 2.2. Then, for each x, y, x , y in E and t in [0, 1], In particular, by letting x = y = z, we obtain U , which implies that U is contractible.
Proof of Proposition 2.3. Let U r := {B r (x, E) : x ∈ X}. By Lemma 2.3, U r is a good cover of B r (X, E). Hence, by the nerve lemma (see [45,Corollary 4G.3]), B r (X, E) is homotopy equivalent to the nerve of U r , which is the same as theČech complexČ r (X, E). By Proposition 2.2,Č r (X, E) = VR 2r (X). This concludes the proof.

Persistence via metric pairs
One of the insights leading to the notion of persistent homology associated to metric spaces was considering neighborhoods of a metric space in a nice (for example Euclidean) embedding [65]. In this section we formalize this idea in a categorical way.
• A metric pair is an ordered pair (X, E) of metric spaces such that X is a metric subspace of E.
• Let (X, E) and (Y, F ) be metric pairs. A 1-Lipschitz map from (X, • Let (X, E) and (Y, F ) be metric pairs and f and g be 1-Lipschitz maps from (X, E) to (Y, F ). We say that f and g are equivalent if there exists a continuous family (h t ) t∈[0,1] of 1-Lipschitz maps from E to F and a 1-Lipschitz map φ : X Y such that h 0 = f, h 1 = g and h t | X = φ for each t.
• We define PMet as the category whose objects are metric pairs and whose morphisms are defined as follows. Given metric pairs (X, E) and (Y, F ), the morphisms from (X, E) to (Y, F ) are equivalence classes of 1-Lipschitz maps from (X, E) to (Y, F ).
Recall the definition of persistence families, Definition 3. We let hTop * denote the category of persistence families with morphisms specified as in Definition 3.
Remark 3.1. Let (X, E) and (Y, F ) be persistent pairs and let f be a 1-Lipschitz morphism between them. Then, f maps B r (X, E) into B r (Y, F ) for each r > 0. Furthermore, if g is equivalent to f , then they reduce to homotopy equivalent maps from B r (X, E) to B r (Y, F ) for each r > 0.
By the remark above, we obtain the following functor from PMet to hTop * .
Definition 15 (Persistence functor). Define the persistence functor B * : PMet hTop * sending (X, E) to the persistence family obtained by the filtration (B r (X, E)) r>0 and sending a morphism between metric pairs to the homotopy classes of maps it induces between the filtrations. Remark 3.2. Suppose a metric pair (X, E) is given. For any k ≥ 0, one can apply the kdimensional homology functor (with coefficients in a given field F) to a persistence family B * (X, E). This induces a persistence module where the morphisms are induced by inclusions. As a persistence module, it is denoted by PH k (B * (X, E); F).
Let Met be the category of metric spaces where morphisms are given by 1-Lipschitz maps. There is a forgetful functor from PMet to Met mapping (X, E) to X and mapping a morphism defined on (X, E) to its restriction to X. Although forgetful functors often have left adjoints, we are going to see that this one has a right adjoint.   Proof. The "only if" part is obvious from Definition 14. Now assume that f | X ≡ g| X . By Lemma 2.2, there exists a geodesic bicombing γ : F × F × [0, 1] F such that for each x, y, x , y in F and t in [0, 1], , t). Note that h 0 = f, h 1 = g and (h t )| X is the same map for all t. The inequality above implies that h t is 1-Lipschitz for all t. This completes the proof. Proof. The uniqueness up to equivalence part follows from Lemma 3.1. The existence part follows from the injectivity of F .

Proof of Theorem 4. Let κ : Met
PMet be the functor sending X to (X, L ∞ (X)) where L ∞ (X) is the Banach space consisting of all the bounded real-valued functions on X with ∞norm (see Definition 12 in Section 2.2). A 1-Lipschitz map f : X Y is sent to the unique morphism (see Lemma 3.2) extending f . This functor κ is said to be the Kuratowski functor.
There is a natural morphsim sending a morphism to its restriction to X. By Lemma 3.2, this is a bijection. Hence κ is a right adjoint to the forgetful functor.
Recall that any two right adjoints of a same functor must be isomorphic [8,Proposition 9.9].
Definition 16 (Metric homotopy pairing). A functor η : Met PMet is called a metric homotopy pairing if it is a right adjoint to the forgetful functor.
Example 3.1. Let (X, d X ) be a metric space. L ∞ (X) is injective space associated to X; see Section 2.2 for the precise definition. Consider also the following additional spaces associated to X: with ∞ -metrics for all of them (cf. [56,Section 3]). Then, (X, L ∞ (X)), (X, E(X)), (X, ∆(X)), (X, ∆ 1 (X)) are all metric homotopy pairings, since the second element in each pair is an injective metric space (see [56,Section 3]) into which X isometrically embeds via the map κ : x d X (x, ·). Here, E(X) is said to be the tight span of X [27,48] and it is a especially interesting space. E(X) is the smallest injective metric space into which X can be embedded and it is unique up to isometry. Furthermore, if X is a tree metric space (i.e., a metric space with 0-hyperbolicity; see Definition 17), then E(X) is the smallest metric tree containing X. This special property has recently been used to the application of phylogenetics, [28].

Isomorphism and stability
Recall that Met is the category of metric spaces with 1-Lipschitz maps as morphisms. We have the functor VR * : Met hTop * induced by the Vietoris-Rips filtration. The main theorem we prove in this section is the following: PMet be a metric homotopy pairing (for example the Kuratowski functor). Then B * • η : Met hTop * is naturally isomorphic to VR 2 * .
Recall the precise definitions of U r andČ r (X, E) from Definition 13. We denote the filtration ofČech complexes Č r (X, E) r>0 byČ * (X, E).
The following theorem is the main tool for the proof of Theorem 5. Its proof, being fairly long, is relegated to Appendix A.3.
Theorem 6 (Generalized Functorial Nerve Lemma). Let X and Y be two paracompact spaces, ρ : X − Y be a continuous map, U = {U α } α∈A and V = {V β } β∈B be good open covers (every nonempty finite intersection is contractible) of X and Y respectively, based on arbitrary index sets A and B, and π : A − B be a map such that ρ(U α ) ⊆ V π(α) for any α ∈ A Let N U and N V be the nerves of U and V, respectively. Observe that, since U α 0 ∩· · ·∩ U αn = ∅ implies V π(α 0 ) ∩ · · · ∩ V π(αn) = ∅, π induces the canonical simplicial mapπ : N U − N V.
Then, there exist homotopy equivalences X − N U and Y − N V that commute with ρ and π up to homotopy: The next corollary is an important special case of Theorem 6. . Let X ⊆ X be two paracompact spaces. Let U = {U α } α∈Λ and U = {U α } α∈Λ be good open covers (every nonempty finite intersection is contractible) of X and X respectively, based on the same index set Λ, such that U α ⊆ U α for all α ∈ Λ. Let N U and N U be the nerves of U and U , respectively. Then, there exist homotopy equivalences X − N U and X − N U that commute with the canonical inclusions X − X and N U − N U , up to homotopy: Proof. Choose the canonical inclusion map X − X as ρ, the identity map on Λ as π, and apply Theorem 6. , which is similar to Theorem 6. The author considers spaces with numerable covers (i.e. the spaces admit locally finite partition of unity subordinate to the covers), whereas in our version that condition is automatically satisfied since we only consider paracompact spaces. Our proof technique differs from that of [74] in that whereas [74] relies on a result from [26], our proof follows the traditional proof of the nerve lemma [45]. such that, for any 0 < r ≤ s, the following diagram commutes up to homotopy: Proof. Observe thatČ r (X, E) is the nerve of the open covering U r for any r > 0, and apply Corollary 4.1.
Furthermore, if we substitute f by an equivalent map, then the homotopy types of the vertical maps remain unchanged.
Proof. Since f is 1-Lipschitz, f (B r (x, E)) ⊆ B r (f (x), F ). Hence, if we choose f | Br(X,E) as ρ, and f | X as π, the commutativity of the diagram is the direct result of Theorem 6.
Furthermore, if f, g are equivalent, then the homotopy (h t ) between f, g induces the homotopy between f r : B r (X, E) − B r (Y, F ) and g r : B r (X, E) − B r (Y, F ). Moreover, since f | X = g| X , both of the induced maps f r :Č r (X, E) − Č r (Y, F ) and g r :Č r (X, E) − Č r (Y, F ) are exactly same.
We are now ready to prove the main theorem of this section.
Proof of Theorem 5. Since all metric homotopy pairings are naturally isomorphic, without loss of generality we can assume that η = κ, the Kuratowski functor. Note that, by Proposition 2.2, C r (X, E) = VR 2r (X) for any X ∈ Met and r > 0.
Let's construct the natural transformation τ from B * • κ : Met hTop * to VR 2 * in the following way: Fix an arbitrary metric space X ∈ Met. Then, let τ X to be the homotopy equivalences φ (X,L ∞ (X)) * : B * (X, L ∞ (X)) − VR 2 * (X) guaranteed by Proposition 4.1. Then, when f : X − Y is 1-Lipschitz, the functoriality between τ X and τ Y is the direct result of Proposition 4.2. So τ is indeed a natural transformation. Finally, since each φ (X,L ∞ (X)) r is a homotopy equivalence for any X ∈ Met and r > 0, τ is natural isomorphism.

Stability of metric homotopy pairings
In this subsection, we consider a distance between metric pairs by invoking the homotopy interleaving distance introduced by Blumberg and Lesnick [11] and then show that metric homotopy pairings are 1-Lipschitz with respect to this distance and the Gromov-Hausdorff distance.
Let us give a quick review of homotopy interleaving distance between R-spaces. For more details, please see [11,Section 3.3]. An R-space is a functor from the poset (R, ≤) to the category of topological spaces. Note that given a metric pair (X, E), the filtration of open neighborhoods B * (X, E) is an R-space. Two R-spaces A * and B * are said to be δ-interleaved for some δ > 0 if there are natural transformations f : A * B * +δ and g : B * A * +δ such that f • g and g • f are equal to the structure maps B * B * +2δ and A * A * +2δ , respectively. A natural transformation f : R * A * is called a weak homotopy equivalence if f induces a isomorphism between homotopy groups at each index. Two R-spaces A * and A * are said to be weakly homotopy equivalent if there exists an R-space R * and weak homotopy equivalences f : R * A * and f : R * A * . The homotopy interleaving distance d HI (A * , B * ) is then defined as the infimal δ > 0 such that there exists δ-interleaved R-spaces A * and B * with the property that A * and B * are weakly homotopy equivalent to A * and B * , respectively.
We now adapt this construction to metric pairs. Given metric pairs (X, E) and (Y, F ), we define the homotopy interleaving distance between them by The main theorem that we are going to prove in this section is the following. Below d GH denotes the Gromov-Hausdorff distance between metric spaces (see [13]) and d I denotes the interleaving distance between persistence modules (see Section 2.1): PMet be a metric homotopy pairing. Then, for any compact metric spaces X and Y , Remark 4.2. Note that through combining Theorem 7 and the isomorphism theorem (Theorem 5), one obtains another proof of Theorem 3: for any compact metric spaces X and Y , a field F, and Proof. Let f : (X, E) (Y, F ) and g : (Y, F ) (X, E) be 1-Lipschitz maps such that f • g and g • f are equivalent to the respective identities. Then, the result follows since f and g induce isomorphism between the R-spaces B * (X, E) and B * (Y, F ).
Proof. By injectivity of E and F , there are 1-Lipschitz maps f : E F and g : F E such that f | X and g| X are equal to id X . Hence, by Lemma 3.1, f • g : (X, F ) (X, F ) and g • f : (X, E) (X, F ) are equivalent to identity.
Proof of Theorem 7. Since all metric homotopy pairings are naturally isomorphic, by Lemma 4.1, without loss of generality we can assume that η = κ, the Kuratowski functor. Let r > d GH (X, Y ).
By assumption (see [13]), there exists a metric space Z containing X and Y such that the Hausdorff distance between X and Y as subspaces of Z is less than or equal to r. Hence, the R-

Application: endpoints of intervals in barc VR k (X)
It is known that, in some cases, the intervals in the Vietoris-Rips barcode of a metric space are of 1. When X is a finite metric space, for any k ≥ 0.
As far as we know the general statement given in Theorem 8 below is first proved in this paper. Our proof crucially exploits the isomorphism theorem (Theorem 5).
Theorem 8. Suppose a compact metric space (X, d X ), a field F, and an nonnegative integer k are given. Then, for any I ∈ barc VR k (X; F), I must be either of the form (u, v] or (u, ∞) for some 0 ≤ u < v < ∞.
The proof of Theorem 8. In this section we first state and prove two lemmas which will be combined in order to furnish the proof of Theorem 8.
Lemma 5.1. Let X be a topological space and G be an abelian group. Then, for any k ≥ 0 and any k-dimensional singular chain c of X with coefficients in G, there exist a compact subset K c ⊆ X and k-dimensional singular chain c of K c with coefficients in G such that where ι : K c − X is the canonical inclusion map.
Proof. Recall that one can express c as a sum of finitely many k-dimensional singular simplices with coefficients in G. In other words, . This K c is the compact subspace that we required.
For the remainder of this section, given any field F and a metric pair (X, E), for each 0 < r < ∞ we will denote by (SC (r) * , ∂ (r) * ) the singular chain complex of B r (X, E) with coefficients in F. For each 0 < r ≤ s < ∞ we will denote by i r,s the canonical inclusion map B r (X, E) ⊆ B s (X, E). By (i r,s ) we will denote the (injective) map induced at the level of singular chain complexes.
Lemma 5.2. Suppose that a compact metric space (X, d X ), a field F, a metric homotopy pairing η, and a nonnegative integer k are given. Then, for every I ∈ barc(PH k (B * • η(X); F)), Proof of (i). Let η(X) = (X, E). The fact that I ∈ barc(PH k (B * • η(X); F)) implies that, for each r ∈ I, there exists a singular k-cycle c r on B r (X, E) with coefficients in F satisfying the following: Now, suppose that u is a closed left endpoint of I (so, u ∈ I). In particular, by the above there exists a singular k-cycle c u on B u (X, E) with coefficients in F with the above two properties.
Then, by Lemma 5.1, we know that there is a compact subset K cu ⊆ B u (X, E) and a singular k-cycle c u on K cu with coefficients in F such that (ι) (c u ) = c u where ι : K cu − B u (X, E) is the canonical inclusion. Moreover, since K cu is compact, there exists small ε > 0 such that is the canonical inclusion. Then, this singular chain satisfies Moreover, c u−ε cannot be null-homologous. Otherwise, there would exist a singular (k + 1)-chain So, we must have [c u−ε ] = 0. But, the existence of such c u−ε contradicts the fact that u is the left endpoint of I. Therefore, one concludes that u cannot be a closed left endpoint, so it must be an open endpoint.
Proof of (ii). Now, suppose that v is an open right endpoint of I (so that v / ∈ I and therefore c v is not defined by the above two conditions). Choose small enough ε > 0 so that v − ε ∈ I, and let Then, c v must be null-homologous.
This means that there exists a singular (k is the canonical inclusion. Then, again by the naturality of boundary operators, but it contradicts the fact that [c v−ε ] = 0. Therefore, v must be a closed endpoint.
Finally, the proof of Theorem 8 follows from the lemmas above.
Proof of Theorem 8. Apply Lemma 5.2 and Theorem 5.
A (false) conjecture. Actually, we first expected the following conjecture to be true. Observe that, if true, the conjecture would imply Theorem 8. Also, it is obvious that this conjecture is true when X is a finite metric space.
Conjecture 1 (Lower semicontinuity of the homotopy type of Vietoris-Rips complexes). Suppose X is a compact metric space. Then, for any r ∈ R >0 , VR r (X) is homotopy equivalent to VR r−ε (X) whenever ε > 0 is small enough.
However, the following example shows that this conjecture is false.
Then, one might now wonder whether the conjecture holds when we restrict the range of r to (0, diam(X)). But, again this new conjecture is false as the following example shows. Example 5.3. Let X := S 1 ∨ α · S 1 for some α ∈ (0, 1). Observe that diam(X) = π. Then, by Lemma 6.2 and Proposition 2.3, VR απ (X) cannot be homotopy equivalent to VR απ−ε (X) for all small enough ε, since on the interval r ∈ [απ − ε, απ], there are infinitly many homotopy types for the Vietoris-Rips complex of the smaller circle.
Remark 5.1. Some time after we discovered the above proof of Theorem 8, we realized that one can actually directly prove the result at the simplicial level which makes the proof slightly simpler; see Appendix A.5.

Application: products and metric gluings
The following statement regarding products of filtrations are obtained at the simplicial level (and in more generality) in [66, Proposition 2.6] and in [38,67]. The statement about metric gluings appeared in [6,Proposition 4] and [62,Proposition 4.4]. These proofs operate at the simplicial level.
Here we give alternative proofs through the consideration of neighborhoods in an injective metric space via Theorem 5.
Theorem 9 (Persistent Künneth Formula). Let X and Y be metric spaces, and F be a field.
(1) (Persistent Künneth formula) Let X × Y denote the ∞ -product of X, Y . Then, (2) Let p and q be points in X and Y respectively. Let X ∨ Y denote the metric gluing of metric spaces X and Y along p and q. Then 2 Remark 6.1. Note that the tensor product of two simple persistence modules corresponding to intervals I and J is the simple persistence module corresponding to the interval I ∩ J. Therefore, the first part of Theorem 9 implies that for any nonnegative integer k.
Example 6.1 (Tori). For a given choice of α 1 , . . . , α n > 0, let X be the ∞ -product Π n i=1 (α i · S 1 ). Then, by [3, Theorem 7.4] and Remark 6.1: Note that above we are defining a multiset, hence if an element appears more than once in the definition, then it will appear more than once in the multiset. In particular, in the case of X = S 1 × S 1 , for all integers k ≥ 0 we have the following: See also the remarks on homotopy types of Vietoris-Rips complexes of tori after [3, Proposition 10.2] and [17].
To be able to prove Theorem 9, we need the following lemmas: Lemma 6.1. If E and F are injective metric spaces, then so is their ∞ -product.
Proof. Let X be a metric space. Note that (f, g) : X E × F is 1-Lipschitz if and only if f and g are 1-Lipschitz. Given such f and g and a metric embedding X into Y , we have 1-Lipschitz extensionsf andg of f and g from Y to E and F , respectively. Hence, (f ,g) : Y E × F is a 1-Lipschitz extension of (f, g). Therefore E × F is injective. Lemma 6.2. If E and F are injective metric spaces, then so is their metric gluing along any two points.
Proof. Let p and q be points in E and F respectively and E ∨ F denote the metric gluing of E and F along p and q. We are going to show that E ∨ F is hyperconvex, hence injective (see Proposition 2.1). We denote the metric on E ∨ F by d, the metric on E by d E and the metric on F by d F . Let Let us show that ≥ 0. If the second element inside the maximum is negative, then there exists Therefore the first element inside the maximum is nonnegative. Hence ≥ 0.
Without loss of generality, let us assume that This implies that the non-empty closed ball where the right-hand side is non-empty by hyperconvexity of F .
Proof of Theorem 9.
(1) Let E and F be injective metric spaces containing X and Y respectively. Let E × F denote the ∞ -product of E and F . Note that for each r > 0, Hence, by the (standard) Künneth formula [63,Theorem 58.5], Now, the result follows from Lemma 6.1 and Theorem 5.
(2) Let E and F be as above and E ∨ F denote metric gluing of E and F along p and q. Note that Hence, by [45,Corollary 2.25], Now, the result follows from Lemma 6.2 and Theorem 5.
7 Application: homotopy types of VR r (X) for X ∈ {S 1 , S 2 , CP n } In a series of papers [50,51,52,53] M. Katz studied the filling radius of spheres and complex projective spaces. In this sequence of papers, Katz developed a notion of Morse theory for the diameter function diam : pow(X) R over a given metric space. By characterizing critical points of the diameter function on each of the spaces S 1 , S 2 , and CP n he was able to prove some results about the different homotopy types attained by B r (X, L ∞ (X)) for X ∈ {S 1 , S 2 , CP n } as r increases. Here, we obtain some corollaries that follow from combining the work of M. Katz [51,52] with Theorem 5.

The case of spheres with geodesic distance
In [46, Theorem 3.5], Hausmann introduced the quantity r(M ) for a Riemannian manifold M , which is the supremum of those r > 0 satisfying the following three conditions: x) < r and w be any point on the shortest geodesic joining x to y.
3. If γ and γ are arc-length parametrized geodesics such that In particular, it can be checked that r(S n ) = π 2 for any n ≥ 1. Hausmann then proved that if r(M ) > 0, VR r (M ) is homotopy equivalent to M for any r ∈ (0, r(M )). This theorem is one of the foundational results in Topological Data Analysis, since it provides theoretical basis for the use of the Vietoris-Rips filtration for recovering the homotopy type of the underlying space.
Then, via Proposition 2.3 we obtain that B r (M, L ∞ (M )) M for r ∈ (0, 1 2 r(M )], and therefore B r (S n , L ∞ (S n )) S n for all r ∈ (0, π 4 ]. In [50, Remark p.508] Katz constructs a retraction from B r (S n , L ∞ (S n )) to S n for r in the range 0, 1 2 arccos −1 n+1 , which is a larger range than the one guaranteed by Hausmann's result. This suggests that an improvement of Hausmann's results might be possible for the particular case of spheres.
Indeed, in the special case of spheres, by a refinement of Hausmann's method of proof (critically relying upon Jung's theorem) we obtain the following theorem which also improves the aforementioned claim by Katz: Theorem 10. For any n ∈ Z >0 , we have VR r (S n ) S n for any r ∈ 0, arccos −1 n+1 .
That this result improves upon Hausmann's follows from the fact that arccos −1 n+1 ≥ π 2 for all integers n ≥ 1. The proof follows from the fact that with the aid of Jung's theorem, one can modify the lemmas that Hausmann originally used. See Appendix A.4 for a detailed proof along these lines which we believe is of independent interest.
Remark 7.1. Note that that the above corollary implies that for every n, barc VR n (S n ; F) contains an interval I n of the form (0, d n ] where d n ≥ arccos −1 n+1 . The corollary does not however guarantee that d n equals its lower bound, nor that I n is the unique interval in barc VR n (S n ; F). Cf.  for an example of a 2-dimensional sphere (with non-round metric) having more than one interval in its 2-dimensional persistence barcode, and see Proposition 9.4 for a general result about I n .
For the particular cases of S 1 and S 2 , we have additional information regarding the homotopy types of their Vietoris-Rips r-complexes when r exceeds the range contemplated in the above Corollary.
The case of S 1 . The complete characterization of the different homotopy types of VR r (S 1 ) as r > 0 grows was obtained by Adamaszek and Adams in [3]. Their proof is combinatorial in nature and takes place at the simplicial level.
Below, by invoking Theorem 5, we show how partial results can be obtained from the work of Katz who directly analyzed the filtration B r (S 1 , L ∞ (S 1 )) r>0 via a Morse theoretic argument.
For each integer k ≥ 1 let λ k := 2πk 2k+1 . Katz proved in [52] that B r (S 1 , L ∞ (S 1 )) changes homotopy type only when r = 1 2 λ k for some k. In particular, his results imply: Hence, the required result follows through Theorem 5.
The case of S 2 . The similar arguments hold for the case of S 2 . Whereas the homotopy types of VR r (S 1 ) for any r > 0 are known [3], we are not aware of similar results for S 2 . Below, E 6 is the binary tetrahedral group.
Proof. B r (S 2 , L ∞ (S 2 )) is homotopy equivalent to the topological join of S 2 and S 3 /E 6 for r ∈  Remark 7.3. As already pointed out in Remark 7.1, by virtue of Theorem 10, (0, d n ] ∈ barc VR n (S n ; F) for some d n ≥ arccos −1 n+1 . Moreover, since for n = 1 and n = 2 we know (by Corollary 7.1 and 7.2) that the homotopy type changes after arccos −1 n+1 , we conclude that barc VR n (S n ; F) contains 0, arccos −1 n+1 for n = 1 and n = 2 and that this is the unique interval in barc VR n (S n ; F) starting at 0. Surprisingly, it is currently unknown how the homotopy type of VR r (S n ) changes after arccos −1 n+1 for n ≥ 3. But, still, in Section 9 we will be able to show that 0, arccos −1 n+1 ∈ barc VR n (S n ; F) for general n via arguments involving the filling radius (cf. Proposition 9.4). In particular, this implies that the homotopy type of VR r (S n ) must change after the critical point r = arccos −1 n+1 since the fundamental class dies after that point, even though we still do not know "how" the homotopy type changes. Moreover, since VR r (S n ) is homotopy equivalent to S n for any r ∈ (0, arccos −1 n+1 ], we know that for any interval I ∈ barc VR n (S n ; F) with I = (0, arccos −1 n+1 ], the left endpoint of I must be greater than or equal to arccos −1 n+1 . The following conjecture is still open except for the n = 1 and n = 2 cases (see [4,Theorem 5.4 and Conjecture 5.7]).

Conjecture 2.
For any n ∈ Z >0 , there exists an ε > 0 such that It is a special case of the following more general homeomorphism: S n * X ∼ = Σ n+1 X for any Hausdorff and locally compact space X. This fact can be proved by induction on n and the associativity of the topological join (see [35,Lecture 2.4]).

The case of CP n
Partial information can be provided for the case of CP n as well. First of all, recall that the complex projective line CP 1 with its canonical metric actually coincides with the sphere S 2 . Hence, one can apply Theorem 10 and Corollary 7.2 to CP 1 . The following results can be derived for general CP n . Corollary 7.3. Let CP n be the complex projective space with sectional curvature between 1/4 and 1 with canonical metric. Then, 1. There exist α n ∈ 0, arccos − 1 3 such that VR r (CP n ) is homotopy equivalent to CP n for any r ∈ (0, α n ].

Let
A be the space of equilateral 4-tuples in projective lines of CP n . Let X be the partial join of A and CP n where x ∈ CP n is joined to a tuple a ∈ A by a line segment if x is contained in the projective line determined by a. There exists a constant β n > 0 such that if then VR r (CP n ) is homotopy equivalent to X.

The case of spheres with ∞ -metric
The Vietoris-Rips filtration of S 1 with the usual geodesic metric is quite challenging to understand [3]. However, it turns out that if we change its underlying metric, the situation becomes very simple. Throughout this section, all metric spaces of interest are embedded in (R n , ∞ ) and are endowed with the restriction of the ambient space metric. In particular, in this section, we use the following conventions: for any n ∈ Z >0 , Note that n ∞ is just the unit closed ∞ -ball around the origin in R n ∞ and n−1 ∞ is its boundary. The following theorem by Kılıç and Koçak is the motivation of this subsection.
Theorem 11 ([55, Theorem 2]). Let X and Y be subspaces of R 2 ∞ . If Y contains X, is closed, geodesically convex 3 , and minimal (with respect to inclusion) with these properties, then Y is the tight span of X.
Theorem 11 has a number of interesting consequences.
Proof. By Theorem 11, the first claim is straightforward. The second claim, namely the explicit is also obvious since we are using the ∞ -norm.
is homotopy equivalent to S 1 for r ∈ (0, 1] and contractible for r > 1. Hence, for any field F, it holds that: Proof. Apply Lemma 7.1 and Theorem 5.
Interestingly, one can also prove the following result.
Proof. By Theorem 11, the first claim is straightforward.
because of the assumptions on z, w, t, and s. Therefore, (z + t, w + s) / ∈ V r so that (z + t, w + s) ∈ D 2 ∞ \V r . By symmetry, the same result holds for other possible sign combinations of z and w.
is homotopy equivalent to S 1 for r ∈ 0, 1 √ 2 and contractible for r > 1 √ 2 . Hence, for any field F, it holds that: Proof. Apply Lemma 7.2 and Theorem 5.
Moreover, it turns out that, despite the fact that Theorem 11 is restricted to subsets of R 2 , Lemma 7.1 can be generalized to arbitrary dimensions.  Proof. When n ≥ 3 one cannot invoke Theorem 11 since it does not hold for general n (see [55,Example 5]). We will instead directly prove that n ∞ is the tight span of n−1 ∞ . First, observe that n Therefore, in order to show that n ∞ is indeed the tight span of n−1 ∞ , it is enough to show that there is no proper hyperconvex subspace of n ∞ containing n−1 ∞ . Suppose this is not true. Then there exists a proper hyperconvex subspace X such that n−1 Without loss of generality, one can assume x 1 ≥ · · · ≥ x n . Now, let and See Figure 4. Then, it is clear that p 0 , p 1 ∈ n−1 ∞ ⊆ X and Therefore, since X is hyperconvex, we know that However, note that This means that p ∈ X, which is contradiction. Hence, no such X exists. Therefore, n ∞ is the tight span of n−1 ∞ , as we required. The second claim, namely the explicit expression of B r ( n−1 ∞ , n ∞ ) is obvious since we are using the ∞ -norm.
Remark 7.5. It seems of interest to study the homotopy types of Vietoris-Rips complexes of ellipsoids with the ∞ -metric, cf. [5].
Here, observant readers would have already noticed that we do not need to use the tight spans of S 1 ∞ and n−1 ∞ in order to apply Theorem 5 since R n ∞ itself is an injective metric space for any n ∈ Z >0 . In particular, the persistent homology of n−1 ∞ is simpler to compute if we use R n ∞ as an ambient space. However, we believe that it is worth clarifying what are the tight spans of S 1 ∞ and n−1 ∞ since the exact shape of tight spans are largely mysterious in general.
We do not know whether D n ∞ is the tight span of S n−1 ∞ for general n. However, if we use R n ∞ as an ambient injective metric space, we are still able to compute its persistent homology.
Theorem 12. For any n ∈ Z >0 and r > 0, In particular, for r > 1 √ n we have V n,r = ∅ so that B r (S n−1 ∞ , D n ∞ ) = D n ∞ . As a result, B r (S n−1 ∞ , R n ∞ ) is homotopy equivalent to S n−1 for r ∈ 0, 1 √ n and contractible for r > 1 √ n . (See Figure 1 for an illustration for the case when n = 2) Observe that it is easy (but, very tedious) to prove that P n is well-defined, continuous, and that P n | S n−1 ∞ = id S n−1 ∞ . Now, for any r > 0, consider the following homotopy The only subtle point is ascertaining whether the image of this map is contained in B r (S n−1 ∞ , R n ∞ ). But, for this note that (x 1 , . . . , x n )−P n (x 1 , . . . , x n ) ∞ < r because of the definition of P n and the fact that (x 1 , . . . , x n ) ∈ B r (S n−1 ∞ , R n ∞ ). Therefore, both (x 1 , . . . , x n ) and P n (x 1 , . . . , x n ) belong to B r (P n (x 1 , . . . , x n ), R n ∞ ) so that the linear interpolation is also contained in B r (P n (x 1 , . . . , x n ), R n ∞ ) ⊂ B r (S n−1 ∞ , R n ∞ ). Hence, one can conclude that B r (S n−1 ∞ , D n ∞ ) is a deformation retract of B r (S n−1 ∞ , R n ∞ ).
Next, let's prove that B r (S n−1 ∞ , D n ∞ ) = D n ∞ \V n,r . First, fix arbitrary (z 1 + t 1 , . . . , z n + t n ) ∈ B r (S n−1 ∞ , D n ∞ ) where n i=1 z 2 i = 1 and t i ∈ (−r, r) for all i = 1, . . . , n. Consider the case of z i ≥ 0 for all i = 1, . . . , n. Then, because of the assumptions on {z i } n i=1 and {t i } n i=1 . Therefore, (z 1 + t 1 , . . . , z n + t n ) / ∈ V n,r so that (z 1 + t 1 , . . . , z n + t n ) ∈ D n ∞ \V n,r . By symmetry, the same result holds for other possible sign combinations of the z i 's. Hence, we have B r (S n−1 ∞ , D n ∞ ) ⊆ D n ∞ \V n,r . Now, fix arbitrary (x 1 . . . , x n ) ∈ D n ∞ \V n,r . Since (x 1 , . . . , x n ) / ∈ V n,r , without loss of generality, one can assume that This completes the proof.
Corollary 7.7. For any n ≥ 2, B r (S n−1 ∞ , R n ∞ ) is homotopy equivalent to S n−1 for r ∈ 0, 1 √ n and contractible for r > 1 √ n . Hence, for any field F, it holds that: Proof. Apply Theorem 12 and Theorem 5.

Application: hyperbolicity and persistence
One can reap benefits from the fact that one can choose any metric homotopy pairing in the statement of Theorem 5, not just the Kuratowski functor.
In this section, we will see one such example which arises from the interplay between the hyperbolicity of the geodesic metric space X and its tight span E(X) (see Example 3.1 to recall the definition of tight span).
We first recall the notion of hyperbolicity.
for all quadruples of points w, x, y, z ∈ X. If a metric space is δ-hyperbolic for some δ ≥ 0, it is said to be hyperbolic. The hyperbolicity hyp(X) of X is defined as the infimal δ ≥ 0 such that X is δ-hyperbolic. A metric space is said to be hyperbolic if hyp(X) is finite.
For a more concrete development on the geometry of hyperbolic metric spaces and its applications (especially to group theory), see [12,43]. 2. All compact Riemannian manifolds are trivially hyperbolic spaces. More interestingly, among unbounded manifolds, Riemannian manifolds with strictly negative sectional curvature are hyperbolic spaces. Observe that "strictly negative" sectional curvature is a necessary condition (for example, consider the Euclidean plane R 2 ).
The following proposition guarantees that the tight span E(X) preserves the hyperbolicity of the underyling space X with controlled distortion. . If X is a δ-hyperbolic geodesic metric space for some δ ≥ 0, then its tight span E(X) is also δ-hyperbolic.
Remark 8.1. Note that since X embeds isometrically into E(X), the above implies that hyp(E(X)) = hyp(X).
The following corollary was already established by Gromov (who attributes it to Rips) in [43, Lemma 1.7.A]. The proof given by Gromov operates at the simplicial level. By invoking Proposition 8.1 we obtain an alternative proof, which instead of operating the simplicial level, exploits the isometric embedding of X into its tight span E(X) (which is a compact contractible space).
Corollary 8.1. If X is a hyperbolic geodesic metric space, then VR 2r (X) is contractible for any r > hyp(X).
By Proposition 2.3, we know that VR 2r (X) is homotopy equivalent to B r (X, E(X)). But, by Proposition 8.1, B r (X, E(X)) = E(X). Since E(X) is contractible by Corollary 2.1, VR 2r (X) is contractible.
As a consequence one can bound the length of intervals in the persistence barcode of hyperbolic spaces.
Observe that metric trees are both 0-hyperbolic and hyperconvex. A recent paper by Joharinad and Jost [49] analyzes the persistent homology of metric spaces satisfying the hyperconvexity condition (which is equivalent to injectivity) as well as that of spaces satisfying a relaxed version of hyperconvexity.
9 Application: the filling radius, spread, and persistence In this section, we recall the notions of spread and filling radius, as well as their relationship. In particular, we prove a number of statements about the filling radius of a closed connected manifold. Moreover, we consider a generalization of the filling radius and also define a strong notion of filling radius which is akin to the so called maximal persistence in the realm of topological data analysis.

Spread
We recall a metric concept called spread. The following definition is a variant of the one given in [50,Lemma 1].
Definition 18 (k-spread). For any integer k ∈ Z >0 , the k-th spread spread k (X) of a metric space (X, d X ) is the infimal r > 0 such that there exists a subset A of X with cardinality at most k such that Finally, the spread of X is defined to be spread(X) := inf k spread k (X), i.e. the set A is allowed to have arbitrary (finite) cardinality. Thus, rad(X) = spread 1 (X). For example, spread(S 1 ) = 2π 3 . Notice that spread(S m ) ≥ spread(S n ) ≥ π 2 for m ≤ n. Katz's proof actually yields that spread n+2 (S n ) = spread(S n ) for each n.

Bounding barcode length via spread
Let (X, d X ) be a compact metric space. Recall that for each integer k ≥ 0, barc VR k (X; F) denotes the persistence barcode associated to PH k (VR * (X); F), the k-th persistent homology induced by the Vietoris-Rips filtration of X (see Section 2.1).
The following lemma is due to Katz [50, Lemma 1].
Lemma 9.1. Let (X, d X ) be a compact metric space. Then, for any δ > spread(X)

2
, there exists a contractible space U such that X ⊆ U ⊆ B δ (X, L ∞ (X)). Remark 9.3. Note that via the isomorphism theorem, Katz's lemma implies the fact that whenever I = (0, v] ∈ barc VR * (X), then v ≤ spread(X). The lemma does not permit bounding the length of intervals whose left endpoint is strictly greater than zero.
It turns out that we can prove a general version of Lemma 9.1 for closed s-thickenings B s (X, L ∞ (X)) for any s ≥ 0.
Note that Lemma 9.1 can be obtained from the case s = 0 in Lemma 9.2. We provide a detailed self-contained proof of this general version in Section 9.2.2.
Armed with Lemma 9.2 and Theorem 5 one immediately obtains item (1) in the proposition below: Proposition 9.1. Let (X, d X ) be a compact metric space, k ≥ 1, and let I be any interval in barc VR k (X; F). Then, and, if I = (u, v] for some 0 < u < v, then v ≤ spread 1 (X).
Remark 9.4. The second part of the proposition above, equation (2), implies that the right endpoint of any interval I (often referred to as the death time of I) cannot exceed the radius rad(X) of X (cf. Remark 9.1).
Note that by [50, Section 1], when X is a geodesic space (e.g. a Riemmanian manifold) we have spread(X) ≤ 2 3 diam(X); this means that we have the following universal bound on the length of intervals in the Vietoris-Rips persistence barcode of a geodesic space X. Corollary 9.1 (Bound on length of bars of geodesic spaces). Let X be a compact geodesic space. Then, for any k ≥ 1 and any I ∈ barc VR k (X; F) it holds that length(I) ≤ 2 3 diam(X).
Remark 9.5. We make the following remarks: • Note that for k = 1, S 1 achieves equality in the corollary above. Indeed, this follows from [3] since the longest interval in barc VR k (S 1 ) corresponds to k = 1 and is exactly (0, 2π 3 ]. • Since VR r (X) is contractible for any r > diam(X), it is clear that length(I) ≤ diam(X) in general. The corollary above improves this bound by a factor of 2 3 when X is geodesic. • In [50] Katz proves that the filling radius of a manifold is bounded above by 1 3 of its diameter. Our result is somewhat more general than Katz's in two senses: his claim applies to Riemannian manifolds M and only provides information about the interval induced by the fundamental class of the manifold (see Proposition 9.4). In contrast, Corollary 9.1 applies to any compact geodesic space and in this case it provides the same upper bound for the length any interval in barc VR k (X; F), for any k.
• Besides the proof via Lemma 9.2 and Theorem 5 explained above, we provide an alternative direct proof of Proposition 9.1 via simplicial arguments. We believe each proof is interesting in its own right.
Proof of Proposition 9.1 via simplicial arguments. Let δ > spread(X). It is enough to show that for each s > 0, the map H k (VR s (X); F) H k (VR δ+s (X); F) induced by the inclusion is zero. By the definition of spread, we know that there is a nonempty finite subset A ⊆ X such that Note that then H k (VR δ (A); F) = 0 because VR δ (A) is a simplex. Let π : X A be a map sending x to a closest point in A. Then, d X (x, π(x)) < δ for any x ∈ X because of the second property of A (moreover, π(x) = x if x ∈ A). Observe that, since diam(π(σ)) < δ for any simplex σ ∈ VR s (X) by the first property of A, this map π induces a simplicial map from VR s (X) to VR δ (A). Hence, one can construct the following composite map ν from VR s (X) to VR δ+s (X): where the second and third maps are induced by the canonical inclusions. Observe that this composition of maps induces a map from H k (VR s (X)) to H k (VR δ+s (X)), and this induced map is actually the zero map since H k (VR δ (A); F) = 0. So, it is enough to show that the composite map ν is contiguous to the canonical inclusion VR s (X) VR δ+s (X). Let σ = {x 0 , . . . , x n } be a subset of X with diameter strictly less than s. Let a i := π(x i ) for i = 0, 1, . . . , n. Then, we have Hence, the diameter of the subspace {x 1 , . . . , x k , a 1 , . . . , a k } is strictly less than δ + s. This shows the desired contiguity and completes the proof. The proof of equation (2) follows similar (but simpler) steps and thus we omit it. Remark 9.6. Note that whereas the proof of Lemma 1 in [50] takes place at the level of L ∞ (X), the proof of Proposition 9.1 given above takes place at the level of simplicial complexes and simplicial maps.

Bounds based on localization of spread
One can improve Proposition 9.1 by considering a localized version of spread. Note that, in [1], the authors also built some bounds on the length of barcodes based on certain notions of size of homology classes.
Definition 19 (Pre-localized spread of a homology class). For each (ω, s) ∈ Spec k (X, F) we define the pre-localized spread of (ω, s) as follows: where S(ω, s) denotes the collection of all B ⊆ X such that ω = ι * ([c]), c is a simplicial k-cycle on VR s (B), and ι : B − X is the canonical inclusion.
Any B as in the definition above will be said to support the homology class (ω, s) ∈ Spec k (X, F).
Example 9.2. In general, I (ω,s) is not necessarily one of the intervals in barc VR k (X; F). Here is a brief sketch of how to construct such an example. Consider the metric graph consisting of 12 vertices and 24 edges as shown in Figure 5. Assume that the length of the edge between adjacent inner (green) vertices is 1, the length of the edge between adjacent outer (orange) vertices is a, and the length of the edge between adjacent inner and outer vertices is b where 1 < a < b < 2. Now, let X be the set of vertices of this graph, and let d X be the shortest path metric between them. Then, one can easily check that barc VR 1 (X; F) = {(1, 2], (a, b]} where (1, 2] is associated to the homology class induced by the inner cycle and (a, b] is associated to the homology class induced by the outer cycle. Now, if we choose {(ω s , s)} s∈(a,b] ⊂ Spec 1 (X, F) corresponds to the interval (a, b] ∈ barc VR 1 (X; F), then I (ωs,s) = (a, 2] / ∈ barc VR 1 (X; F) for s ∈ (a, b]. Definition 20 (Localized spread of a homology class). For each (ω, s) ∈ Spec k (X, F), we define the localized spread of (ω, s) as follows: spread(X; ω, s) := sup{pspread(X; ω , s ) : s ≤ s and ω = (i s ,s ) * (ω )}. Remark 9.8. It is easy to check that both pspread(X; ω, s) and spread(X; ω, s) are always upper bounded by spread(X).
The following Proposition 9.2 is the "localized" version of Proposition 9.1 we promised in the beginning of this section. Proposition 9.2. Let (X, d X ) be a compact metric space and k ≥ 1. Then, for any (ω, s) ∈ Spec k (X, F), we have length(I (ω,s) ) ≤ spread(X; ω, s).
For an arbitrary I ∈ barc VR k (X; F), a family of nonzero homology classes {(ω s , s)} s∈I ⊆ Spec k (X, F) such that (i s,s ) * (ω s ) = ω s for any s ≤ s in I where i s,s : VR s (X) − VR s (X) is the canonical inclusion, will be said to correspond to I, if there is an isomorphism Observe that Theorem 1 guarantees that at least one such family of nonzero homology classes {(ω s , s)} s∈I always exists. Remark 9.9. Now, given an arbitrary I ∈ barc VR k (X; F), there is a family of nonzero homology classes {(ω s , s)} s∈I ⊆ Spec k (X, F) corresponding to I as described above. Then, obviously I ⊆ I (ωs,s) for each s ∈ I. Hence, length(I) ≤ inf s∈I length(I (ωs,s) ) ≤ inf s∈I spread(X; ω s , s) ≤ spread(X) so that one recovers the result in Proposition 9.1. Below we show some examples that highlight cases in which the localized spread is more efficient at estimating the length of bars than its global counterpart. Let X be a compact metric space. If for a given I ∈ barc VR k (X; F) a corresponding family {(ω s , s)} s∈I ⊆ Spec k (X, F) is supported by a subset B ⊆ X, then where the first inequality holds as in Remark 9.9, the second inequality holds by Proposition 9.2, and the last inequality follows from Remark 9.8.
Here are 3 scenarios in which the estimate in inequality (3)  2. Let X be the metric gluing of a loop of length l 2 and an interval of length l 1 (glued at one of its endpoints). Then, by Proposition 9.1, I ≤ spread(X) for any I ∈ barc VR k (X; F). However, observe that one can make spread(X) arbitrarily large by increasing l 1 . But, if J ∈ barc VR 1 (X; F) and a family of nonzero homology classes {(ω s , s)} s∈J ⊆ Spec 1 (X, F) corresponding to J is supported by the loop, then length(J) ≤ spread(B) = l 2 3 by the inequality (3) and Remark 9.2. Again, as above example, J = 0, l 2 3 .
3. An example similar to the one described in the previous item arises from Figure 3. Consider the tube connecting the two blobs to be large: in that case the standard spread of the space will be large yet the lifetime of the individual H 2 classes will be much smaller.

The proof of Lemma 9.2
Let us introduce a technical tool for this subsection. It is easy to check that the usual linear interpolation in L ∞ (X) gives a geodesic bicombing on L ∞ (X) satisfying all three properties mentioned in Lemma 2.2. However, in [50], Katz introduced an alternative way to construct a geodesic bicombing on L ∞ (X): Definition 21 (Katz's geodesic bicombing). Let X be a compact metric space. We define the Katz geodesic bicombing γ K on L ∞ (X) in the following way: where In other words, γ K (f, g, ·) moves from f to g with the same speed at every point.
The following proposition establishes that γ K is indeed a (continuous) geodesic bicombing, amongst other properties. The proof is relegated to Appendix A.1. Proposition 9.3. Let X be a compact metric space. Then, the Katz geodesic bicombing γ K on L ∞ (X) satisfies the following properties: for any f, g, h ∈ L ∞ (X) and 0 ≤ s ≤ t ≤ 1, 1]. (This property is called consistency).
Properties (2) Proof of Lemma 9.2. By the definition of spread, we know that there is a nonempty finite subset A ⊆ X and δ ∈ (0, δ) such that diam(A) < 2δ and sup x∈X inf a∈A d X (x, a) < 2δ .
Next, we define The main strategy of the proof is depicted in Figure 6.
Proof of Claim 9.1. To prove this, fix arbitrary x ∈ X. Note that Since Also, because the diameter of A is smaller than 2δ , we have d X (a, x) − d X (x, A) − δ < δ . Therefore, we have |d X (a, x) − f (x)| ≤ δ . Furthermore, if we put x = a, we have that d X (a, ·) − f ∞ = δ . Figure 6: Strategy of the proof of Lemma 9.2. By construction, the distance between f and d X (a, ·) (represented by the red dot) is δ and the distance between γ K (g, f, t 0 ) and d X (a, ·) is less than or equal to s + δ . Hence, by Item (6) of Proposition 9.3, we have that the point represented by a square will be at distance at most s + δ from d X (a, ·).
Then U s,δ obviously contains B s (X, L ∞ (X)) and can be contracted to the point f . The lemma will follow once we establish the following claim.
Proof of Claim 9.2. To see this, fix arbitrary g ∈ B s (X, L ∞ (X)) and t ∈ [0, 1]. Note that one can choose x ∈ X such that g − d X (x, ·) ∞ ≤ s.
Fix arbitrary x ∈ X. If |g(x ) − f (x )| ≤ δ , then γ K (g, f, t 0 )(x ) = f (x ). Hence, cannot be between d X (a, x ) and f (x ) since |d X (a, x ) − f (x )| ≤ δ by Claim 9.1. This implies that either Either way, it is easy to see that we always have where the last inequality is true because Therefore, combining this inequality with Claim 9.1 and property (6) of Proposition 9.3, one finally obtains that This concludes the proof.

The filling radius and Vietoris-Rips persistent homology
Now, we recall the notion of filling radius, an invariant for closed connected manifolds introduced by Gromov [42, pg.8] in the course of proving the systolic inequality (see also [54] for a comprehensive treatment). It turns out to be that this notion can be a bridge between topological data analysis and differential geometry/topology.  Remark 9.11 (Relative filling radius and minimality for injective metric spaces.). Note that the relative filling radius can be defined for every metric pair (M, E) by considering r-neighborhoods of M in E -it is denoted by FillRad(M, E). Gromov [42] showed that we obtain the minimal possible relative filling radius through the Kuratowski embedding (that is when E = L ∞ (M )). This also follows from our work but in greater generality in the context of embeddings into injective metric spaces. If M can be isometrically embedded into an injective metric space F , then this embedding can be extended to a 1-Lipschitz map f : E F , which induces a map of filtrations  is not homotopy equivalent to CP 3 for r ∈ 1 2 arccos − 1 3 , 1 2 arccos − 1 3 + ε 0 where ε 0 > 0 is a positive constant. In other words, the homotopy type of B r (CP 3 , L ∞ (CP 3 )) already changed before r = FillRad(CP 3 ).

2.
The following example provides geometric intuition for how the homotopy type of Kuratowski neighborhoods may change before r reaches the filling radius. Consider a big sphere with a small handle attached through a long neck (see Figure 7). Since the top dimensional hole in this space is big, we expect the filling radius to be big. On the other hand, the 1-dimensional homology class coming from the small handle dies in a small Kuratowski neighborhood, hence the homotopy type changes at that point.
We now relate the filling radius of a closed connected n-dimensional manifold to its n-dimensional Vietoris-Rips persistence barcode.  In this case, as r > 0 increases, B r (X, L ∞ (X)) changes homotopy type from that of X to that of S 2 as soon as r > r 0 for some r 0 < FillRad(X).
The unique interval identified by Proposition 9.4 will be henceforth denoted by Proof of Proposition 9.4. First, let us consider the case when M is orientable. Observe that the following diagram commutes: The proof of the non-orientable case is similar, so we omit it. Remark 9.13. Actually, one can generalize Proposition 9.4 to metric manifolds. See Proposition 9.7 for full generalization. A similar result is true for the ∞ -product of more than two metric manifolds. for any closed Riemannian manifold M . Observe that the n-dimensional torus T n is an aspherical, hence essential, manifold. Also, observe that sys 1 (T n ) = sysh 1 (T n ) since the fundamental group π 1 (T n ) is abelian. Therefore, this permits relating the top dimensional persistence barcode with the first dimensional barcode of any n-dimensional Riemannian torus. We summarize this via the following Corollary 9.2. For any Riemannian metric on the n-dimensional torus T n : Figure 8: A space X for which sys(X) = sysh(X).
• the interval I T n 1 := 0, sys 1 (T n ) 3 is an element of barc VR 1 (T n ; F), • the interval I T n n := (0, 2 FillRad(T n )] is an element of barc VR n (T n ; F), and Volume and persistence barcodes.
It then follows that d M n ≤ 2 c n vol(M ) 1/n .
In particular, this bound improves upon the one given by Corollary 9.1, d M n ≤ 2 3 diam(M ), when M is "thin" like in the case of a thickened embedded graph [61].
Spread and persistence barcodes. The following proposition is proved in [50, Lemma 1]. Here we provide a different proof which easily follows from the persistent homology perspective that we have adopted in this paper. Proof. Follows from Proposition 9.1 and Proposition 9.4.
Remark 9.15. One can also use Lemma 9.1 to prove Proposition 9.5.
Remark 9.16. The inequality in the statement above becomes an equality for spheres [50].
By Corollary 9.1, Proposition 9.4, and the fact that FillRad(S 1 ) = π 3 , we know that for any k ≥ 1, and any I ∈ barc VR k (S 1 ; F). This motivates the following conjecture: for any I ∈ barc VR k (M ; F) and any k ≥ 1.
However, this conjecture is not true in general, as the following example shows.

Application to obtaining lower bounds for the Gromov-Hausdorff distance
With the aid of the stability of barcodes (cf. Theorem 3) and the notion of filling radius, one can obtain the following result: Proposition 9.6. Let M be a closed connected m-dimensional orientable (resp. non-orientable) Riemannian manifold, and let X be a compact metric space such that: 1. H m (X; F) = 0 for some arbitrary field F (resp. H m (X; F) = 0 for F = Z 2 ), and 2. VR r (X) X for every r ∈ (0, FillRad(M )]. Then, Proof. Observe that by Theorem 2 and Theorem 3, Hence, it is enough to establish that Proof. Observe that S m is orientable, H m (S n ; F) = 0 for any field F, VR r (S n ) S n for any r ∈ 0, arccos −1 n+1 by Theorem 10, and arccos −1 by Proposition 9.6. This, taken together with the comments above leads to the following conjecture.
Conjecture 4. For any geodesic, compact, and simply connected space Y we have

A generalization of the filling radius
The goal of this section is to provide some partial results regarding the structure of barc VR * (·) for non-smooth spaces; see Figure 9. In order to do so we consider a generalization of the notion of filling radius for arbitrary compact ANR metric spaces and arbitrary homology dimension. See [47] for an introduction to the general theory of ANRs.
Definition 23 (Absolute neighborhood retract). A metric space (X, d X ) is said to be ANR (Absolute Neighborhood Retract) if, whenever X is a subspace of another metric space Y , there exists an open set X ⊂ U ⊆ Y such that X is a retract of U . It is known that every topological manifold with compatible metric (so, a metric manifold) is an ANR. Not only that, every locally Euclidean metric space is an ANR (see [47,Theorem III.8.1]). Also, every compact, (topologically) finite dimensional, and locally contractible metric space is ANR (see [30, Section 1]). The following example is one application of this fact.
Example 9.4. Let G be a compact metric graph and M 1 , . . . , M n be closed connected metric manifolds. Choose points v 1 , . . . , v n ∈ G and p i ∈ M i for each i = 1, . . . , n and consider the geodesic metric space X := G ∨ M 1 ∨ · · · ∨ M n arising from metric gluings via v 1 ∼ p 1 , . . . , v n ∼ p n . Since X is compact, (topologically) finite dimensional, and locally contractible, it is an ANR. See Figure 9.
Finally, we are ready to define a generalized filling radius.
Definition 24 (Generalized filling radius). Let (X, E) be a metric pair where X is a compact ANR metric space. For any integer k ≥ 1, any abelian group G, and any ω ∈ H k (X; G), we define the generalized filling radius as follows: where ι E r : X B r (X, E) is the (corestriction of the) isometric embedding. In other words, we have the map FillRad k ((X, E), G, ·) : H k (X; G) − R ≥0 . Remark 9.19. Following the discussion in Remark 9.11 after Equation (4), one can also prove that the smallest possible value of the generalized filling radius is attained when E is an injective metric space. Hence, we denote FillRad k (X, G, ω) instead of FillRad k ((X, E), G, ω) whenever E is injective, for simplicity.
Let M be an n-dimensional metric manifold. Then, note that we have FillRad n (M, G, [M ]) = FillRad(M ) in the following two cases: (1) when M is orientable and G = Z, and (2) when M is non-orientable and G = Z 2 .
A priori, one can define the generalized filling radius for any metric space X. However, we believe that the context of ANR metric spaces is the right level of generalization for our purposes because of the following proposition analogous to Proposition 9.4. Proposition 9.7. Let X be a compact ANR metric space. Then, for any k ≥ 1 and nonzero ω ∈ H k (X; G), we have FillRad k (X, G, ω) > 0, and (0, 2 FillRad k (X, G, ω)] ∈ barc VR k (X ; F) where (1) G = Z and F is an arbitrary field, or (2) G = F is an arbitrary field.
Proof. First, note that one cannot apply Hausmann's theorem since X is not necessarily a Riemannian manifold. However, since X is ANR and a closed subset of L ∞ (X), there exists an open U ⊆ L ∞ (X) such that X ⊂ U and U retracts onto X. Let ρ : U X be the retraction. Now, since U is open there exists r > 0 such that B r (X, L ∞ (X)) ⊆ U . Observe that the restriction ρ r := ρ| Br(X,L ∞ (X)) : B r (X, L ∞ (X)) X is still a retraction. It means that ρ r • ι r = id X . Therefore, is injective. This implies that FillRad k (X, G, ω) > 0 and that there exists some interval in barc VR k (X; F) corresponding to the nonzero homology class ω ∈ H k (X; G). The remaining part of proof is essentially the same as the proof of Proposition 9.4, so we omit it. where γ is a shortest closed curve representing the homology class ω.
A refinement for the case k = 1. We now prove that when k = 1, the intervals given by Proposition 9.7 are the only bars in barc VR 1 (X; F).
Lemma 9.4. Let X be a compact geodesic metric space, which is a subspace of an injective metric space (E, d E ). Then, for any r > 0, the canonical inclusion ι r : X − B r (X, E) induces a surjection at the level of fundamental groups. In particular, this implies ι r induces a surjection at the level of first degree of homology, also.
Proof. Let γ : [0, 1] B r (X, E) be an arbitrary continuous path with endpoints x, x in X. It is enough to show that γ is homotopy equivalent to a path in X relative to its endpoints. By Lebesgue number lemma, one can choose 0 = t 0 < t 1 < · · · < t n = 1 such that there exists is homotopy equivalent to α i · β i relative to endpoints. Hence we have γ (α 1 * β 1 ) * · · · * (α n * β n ) relative to endpoints. Note that α 1 and β n can be chosen as geodesics in X as they connect x, x 1 and x n , x in B r (x 1 , E), B r (x n , E) respectively. Hence it is enough to show that (β 1 * α 2 ) * · · · * (β n−1 * α n ) is homotopy equivalent to a path in X relative to endpoints. Let us show that β i · α i+1 is homotopy equivalent to a path in X for each i. Let p be a midpoint of x i , x i+1 in X. Note that p and y i are contained in B r (x i , E) ∩ B r (x i+1 , E), which is contractible (again by Lemma 2.3). Let θ be a path in that intersection from y i to p. Let γ x i ,p be a shortest geodesic in X from x i to p and γ p,x i+1 be a shortest geodesic in X from p to x i+1 . Note that γ x i ,p ·θ is contained in B r (x i ) and has endpoints x i , y i hence it is homotopy equivalent to β i relative to endpoints. Similarly θ · γ p,x i+1 is homotopy equivalent to α i+1 relative to endpoints. Hence relative to endpoints. This completes the proof of the first claim. In [73,Theorem 8.10], Z. Virk provided a proof of the Corollary below which takes place at the simplicial level. The proof we give below exploits the hyperconvexity properties of L ∞ (X) and also our isomophism theorem, Theorem 5. Our proof is much more concise. See [25, Section 3] for related results.
A conjecture. After seeing the proof of Proposition 9.7, some readers might wonder whether one can prove a version of Hausmann's theorem [46,Theorem 3.5] for compact ANR metric spaces. This leads to formulating the conjecture below.
Conjecture 5. Let (X, d X ) be a compact ANR metric space. Then, there exists r(X) > 0 such that VR r (X) is homotopy equivalent to X for any r ∈ (0, r(X)].

Rigidity of spheres
A problem of interest in the area of persistent homology is that of deciding how much information from a metric space is captured by its associated persistent homology invariants. One basic (admittedly imprecise) question is: Question 1. Assume X and Y are compact metric spaces such that barc VR k (X; F) = barc VR k (Y ; F) for all k ∈ Z ≥0 . Then, how similar are X and Y (in a suitable sense)?
As proved in [62] via the notion of core of a metric graph, the unit circle S 1 and the join X of S 1 with disjoint trees of arbitrary length (regarded as a geodesic metric space) have the same Vietoris-Rips persistence barcodes (for all dimensions), see Figure 10. However, by increasing the length of the trees attached these two spaces are at arbitrarily large Gromov-Hausdorff distance, as shown in Figure 10. This means that, in full generality, Question 1 does not admit a reasonable answer if "similarity" is measure in a strict metric sense via the Gromov-Hausdorff distance.
A related type of questions one might pose are of the type: Figure 10: Two geodesic spaces with the same Vietoris-Rips persistence barcodes. Notice that these spaces are at a large Gromov-Hausdorff distance.
Question 2. Let C be a given class of compact metric spaces. Does there exist C > 0 such that whenever d B (barc VR * (X), barc VR * (Y )) < C for some X, Y ∈ C, then X and Y are homotopy equivalent?
Answers to questions such as the two above are not currently known in full generality. One might then consider "localized" versions of the above questions: fix some special compact metric space X 0 , and then assume Y satisfies the respective conditions stipulated in the above question statements.
In this regard, from work by Wilhelm [75, Main Theorem 2] and Proposition 9.4 we immediately obtain the following corollary for the case of Riemannian manifolds. 2. If I M n = I S n n then M is isometric to S n .
3. There exists n > 0 such that if length(I S n n ) − n < length(I M n ), then M is diffeomorphic to S n . 4. If length(I M n ) > π 3 , then M is a twisted n-sphere (and in particular, homotopy equivalent to the n-sphere). Remark 9.20. Note that the case of n = 1 is simpler. Let M be an arbitrary closed connected 1-dimensional Riemannian manifold. Then, M is isometric to r · S 1 for some r > 0 and I M 1 = 0, 2πr 3 . Hence, I M 1 = I S 1 1 obviously implies M is isometric to S 1 .
Remark 9.21. Wilhelm's method of proof does not yield an explicit value for the parameter n given in item 2 above. Wilhelm's rigidity result was extended to Alexandrov spaces by Yokota [76], so Corollary 9.5 can be generalized to that context. Example 9.6 (A one parameter family of surfaces with the same Filling Radius as S 2 ). If we ignore the sectional curvature condition in Corollary 9.5, then for each ε > 0 small enough one can construct a one-parameter family {S 2 h : h ∈ [0, FillRad(S 2 ) − ε]} of surfaces with the same filling radius as S 2 such that S 2 0 = S 2 but S 2 h is not isometric to S 2 for any h > 0. This phenomenon is analogous to the one depicted in Figure 10.
Here is the construction (cf. Figure 2): Let u 1 , u 2 , u 3 , u 4 be the vertices of a regular tetrahedron inscribed in S 2 . Hence, d S 2 (u i , u j ) = 2 FillRad(S 2 ) for any i = j. Now, let T be a very small spherical triangle contained inside the spherical triangle determined by the points u 1 , u 2 , u 3 as in Figure 2 (left). In other words, we choose ε := diam(T ) 2 FillRad(S 2 ). Now, for any h ≥ 0, we define S 2 h in the following way: h is a 2-dimensional metric manifold. See Figure 2 (right) for the description of S 2 h . Also, note that the following map is 1-Lipschitz.
for any h ≥ 0. Proof. Note that, since P h is 1-Lipschitz, the following diagram commutes for any r > 0.
, the diagram above implies FillRad(S 2 h ) ≥ FillRad(S 2 ) as we required. Claim 9.4. Next, we claim that FillRad(S 2 h ) ≤ FillRad(S 2 ) whenever h + ε ≤ 2 FillRad(S 2 ). Proof. For this we will prove that the spread of S 2 h is bounded by twice the filling radius of S 2 . Note that the set {(u 1 , 0), (u 2 , 0), (u 3 , 0), (u 4 , 0)} ⊂ S 2 h satisfies the following conditions, Observe that the second condition holds because if (x, s) ∈ ∂T ×[0, h]∪T ×{h} (the triangular cylinder with its cap), We then conclude that FillRad(S 2 ) = FillRad(S 2 h ) whenever h ∈ [0, 2 FillRad(S 2 ) − ε]. Remark 9.22. Note that the above construction can be generalized to S n for n ≥ 3. Also, the small subset T is need not be a spherical triangle in general, though the argument becomes more involved in that case. For example, one can choose T to be a small geodesic disk on S 2 . barc VR n (S n ; F) other than (0, 2FillRad(M )] must be born after 2FillRad(S n ). In particular, u ≥ 2FillRad(S n ). This implies which is contradiction. Hence, (0, 2FillRad(M )] is matched to (0, 2FillRad(S n )] in the optimal matching. Therefore, The proof strategy for Propositions 9.8 and 9.9 is to invoke Wilhelm's result [75,Main Theorem 2] and Lemma 9.5 above. However, if FillRad(M ) were small, one would not be able to apply Wilhelm's theorem. To avoid that, we will invoke a result due to Liu [59].

Stability of the filling radius
In [59], Liu studies the mapping properties of the filling radius. His results can be interpreted as providing certain guarantees for how the filling radius changes under multiplicative distortion of metrics. Here we study the effect of additive distortion.
This question is whether the filling radius could be stable as a map from the collection of all metric manifolds to the real line. The answer is negative, as the following example proves and, by Remark 9.14, FillRad(N ) = π 3 . This means that equation (8) cannot hold in general.
A subsequent possibility is considering only manifolds with the same dimension. The answer in this case is also negative: Example 9.8 (Counterexample for manifolds with the same dimension). Let n ≥ 2 be any integer and , δ > 0; we assume that δ so that a certain tubular neighborhood construction described below works. Consider M = S n ⊂ R n+1 . Endow S n with the usual round Riemannian metric. Let G be a (finite) metric graph embedded in S n such that d GH (S n , G ) < ; such graphs always exist for compact geodesic spaces [13,Proposition 7.5.5]. Now, let N ,δ be (a suitably smoothed out version of) the boundary of the δ-tubular neighborhood of G in R n+1 . Now, d GH (M, N ,δ ) ≤ C · ( + δ), for some constant C > 0 whose exact value is not relevant. However, FillRad(M ) = 1 2 arccos −1 n+1 ≥ π 4 whereas FillRad(N ,δ ) ≤ C n · δ by inequality (6). This means that equation (8) cannot hold in general, even when the manifolds M and N have the same dimension.
We are however able to establish the following: Actually, one can prove a more general result.

Now, fix arbitrary r >
be the canonical inclusion map. The maps defined above fit into the following (in general noncommutative diagram): Next, we prove that j r • i r 1 is homotopic to i via the linear interpolation The only subtle point is whether this linear interpolation is always contained in the thickening or not. To ascertain this, for arbitrary x ∈ M and t ∈ [0, 1], compute the distance between H(x, t) and d 2 (x, ·) as: Hence, H is a well-defined homotopy between j r • i r 1 and i . Therefore, we have From the assumption on r, we know that (i r 1 ) * (ω) = 0. By the above, this implies that Hence, we have d 1 ), G, ω) is arbitrary. In the similar way, one can also show This concludes the proof.

The strong filling radius
Examples 9.7 and 9.8 suggest that the setting of Proposition 9.10 might be a suitable one for studying stability of the filling radius. In this section we consider a certain strong variant of the filling radius satisfying equation (8)  The reader familiar with concepts from applied algebraic topology will have noticed that the definition of strong filling radius of an n-dimensional metric manifold coincides with (one half of) the maximal persistence of its associated Vietoris-Rips persistence module. In fact, for each nonnegative integer k one can define the k-dimensional version of strong filling radius of any compact metric space X.
Definition 26 (Generalized strong filling radius). Given a compact metric space X, a field F, and a nonnegative integer k ≥ 0, we define the generalized strong filling radius sFillRad k (X; F) as half the length of the largest interval in the k-th Vietoris-Rips persistence barcode of X: sFillRad k (X; F) := 1 2 max length(I), I ∈ barc VR k (X; F) .

Remark 9.24.
• When X is isometric to a metric manifold M with dimension n, we of course have sFillRad n (X) = sFillRad(M ).
• In general, sFillRad k and FillRad k are obviously related in the sense that sFillRad k (X; F) ≥ sup{FillRad k (X, F, ω); ω ∈ H k (X; F)} for any nonnegative integer k.
The following remark follows directly from Proposition 9.1 and Proposition 9.4. Remark 9.26. For each n ≥ 1, the n-dimensional unit sphere with the intrinsic metric is Fregularly filled for any field F. Indeed, by [50, Proof of Theorem 2], FillRad(S n ) = 1 2 spread(S n ). Hence, the result follows from Remark 9.25.
As a consequence of the remark above and Remark 9.2 we have Corollary 9.6. For all integers n ≥ 1, FillRad(S n ) = sFillRad(S n ; F) = 1 2 arccos −1 n+1 . There exist, however, non-regularly filled metric manifolds. We present two examples: the first one arises from our study of the Künneth formula in Section 6 whereas the second one is a direct construction. Both examples make use of results from [3] about homotopy types of Vietoris-Rips complexes of S 1 .
Example 9.9 (A non-regularly filled metric manifold). Let r > 1 and X be the ∞ -product S 1 × S 1 × (r · S 1 ). By Remark 9.14, FillRad(X) = FillRad(S 1 ) = 2π 3 . By Example 6.1, barc VR 3 (X; F) contains the interval ( 2πr 3 , 4πr 5 ], which has length 2πr 15 . Hence, if r > 5, X is not F-regularly filled. Example 9.10 (A non-regularly filled Riemannian manifold). Take any embedding of S 1 into R 4 and let > 0 be small. Consider the boundary C of the -tubular neighborhood around S 1 . This will be a 3-dimensional submanifold of R 4 . As a submanifold it inherits the ambient inner product and C can be regarded as a Riemannian manifold in itself. Then, as a metric space, with the geodesic distance, C will be -close to S 1 (with geodesic distance) in Gromov-Hausdorff sense. Because we know that for r ∈ ( 2π 3 , 4π 5 ], VR r (S 1 ) S 3 , and because of the Gromov-Hausdorff stability of barcodes (cf. Theorem 3), it must be that barc VR 3 (C ; F) contains an interval I which itself contains ( 2π 3 + , 4π 5 − ). This latter interval is non-empty whenever > 0 is small enough so that sFillRad(C ; F) ≈ 2π 15 − 2 . However, FillRad(C ) ≈ .
By invoking the relationship between the Vietoris-Rips persistent homology and the strong filling radius, one can verify that the strong filling radii of two n-dimensional metric manifolds M and N are close if these two manifolds are similar in the Gromov-Hausdorff distance sense.
Proposition 9.12. Let X and Y be compact metric spaces . Then, for any integer k ≥ 0,

Proof. By Remark 4.2 one has
where the last inequality follows from the triangle inequality for the interleaving distance. The conclusion now follows from Example 2.1.
Remark 9.27. Albeit for the notation sFillRad k , the above stability result should be well known to readers familiar with applied algebraic topology concepts -we state and prove it here however to provide some background for those readers who are not.
(3) Fix arbitrary x ∈ X. We will prove that Unfortunately, we have to do tedious case-by-case analysis.

Hence
, In the similar way, one can also obtain for any t ∈ [0, 1]. Hence, as we wanted.
: This case is similar to the previous one so we omit it.
(c) If f (x) ≥ g(x) and h(x) ≤ g(x): Observe that Therefore, and h(x) ≥ g(x): Similar to the previous case.
Since x is arbitrary, we finally have (4) Fix arbitrary x ∈ X. We will prove that Let's do case-by-case analysis.

Hence
, In the similar way, one can also obtain for any t ∈ [0, 1]. Hence, as we wanted.
(b) If f (x) ≤ g(x) and f (x) ≤ h(x): Similar to the previous case. Therefore, and f (x) ≥ h(x): Similar to the previous case.
Since x is arbitrary, we finally have By property (1) of this proposition, we know Observe that, One can do the similar proof for the case when f (x) ≤ g(x). Hence, we have γ K (φ, ψ, λ) = γ K (f, g, (1 − λ)s + λt).
(6) Consider the special case s = 0 and t = 1. Fix arbitrary x ∈ X. Observe that γ K (f, g, r)(x) is between f (x) and g(x). Therefore, Since x is arbitrary, we have For general s and t, combine this result with property (5).

A.3 Proof of the generalized functorial nerve lemma
The goal of this appendix is proving the following Generalized Functorial Nerve Lemma.
Theorem 6 (Generalized Functorial Nerve Lemma). Let X and Y be two paracompact spaces, ρ : X − Y be a continuous map, U = {U α } α∈A and V = {V β } β∈B be good open covers (every nonempty finite intersection is contractible) of X and Y respectively, based on arbitrary index sets A and B, and π : A − B be a map such that ρ(U α ) ⊆ V π(α) for any α ∈ A Let N U and N V be the nerves of U and V, respectively. Observe that, since U α 0 ∩· · ·∩ U αn = ∅ implies V π(α 0 ) ∩ · · · ∩ V π(αn) = ∅, π induces the canonical simplicial mapπ : N U − N V.
Then, there exist homotopy equivalences X − N U and Y − N V that commute with ρ and π up to homotopy: Our proof of Theorem 6 invokes many elements of [45,Section 4.G] which provides a proof of the classical nerve lemma.
Then, the complex of spaces corresponding to U consists of the set of all U σ and the set of all canonical inclusions i σσ over all possible σ ⊆ σ ∈ N U.
The realization of this complex of spaces, denoted by ∆X U , is defined in the following way: where (x, p) ∼ (x , p ) whenever i σσ (x) = x and j σσ (p ) = p.
Proposition A.1 (Proposition 4G.1 of [45]). Let X be a topological space and U = {U α } α∈Λ be a good open cover of X (every nonempty finite intersection is contractible). Then, is a homotopy equivalence between ∆X U and N U.
Proof. First of all, since U σ is contractible whenever σ ∈ N U, note that there is a homotopy equivalence φ σ : U σ { * } for any σ ∈ N U. The homotopy equivalence between ∆X U and N U is just a special case of [45,Proposition 4G.1]. The choice of f is implicit in the fact that both of ∆X U and N U are deformation retracts of ∆M X U where ∆M X U is the realization of the complex of spaces consisting of the mapping cylinders M φ σ for any σ ∈ N U and the canonical inclusions between them.
Proposition A.2 (Proposition 4G.2 of [45]). Let X be a paracompact space, U = {U α } α∈Λ be an open cover of X, and {ψ α } α∈Λ be a partition of unity subordinate to the cover U (it must exist since X is paracompact). Then, is a homotopy equivalence between X and ∆X U .
Lemma A.1. Let X and Y be two topological spaces, ρ : X − Y be a continuous map, U = {U α } α∈A and V = {V β } β∈B be good open covers (every nonempty finite intersection is contractible) of X and Y respectively, based on arbitrary index sets A and B, and π : A − B be a map such that for any α ∈ A. Let N U and N V be the nerves of U and V, respectively. Observe that π induces the canonical simplicial mapπ : N U − N V since U α 0 ∩ · · · ∩ U αn = ∅ implies V π(α 0 ) ∩ · · · ∩ V π(αn) = ∅, and ρ induces the canonical mapρ : ∆X U − ∆Y V mapping (x, p) to (ρ(x),π(p)).
Then, there exist homotopy equivalences f : ∆X U − N U and f : ∆Y V − N V which commute withρ andπ: is a homotopy equivalence between ∆X U and N U. Also, is a homotopy equivalence between ∆Y V and N V.
To check the commutativity of the diagram, fix arbitrary (x, p) ∈ U σ × ∆ n ⊆ ∆X U . Then, Hence,π • f = f •ρ as we wanted.
Lemma A.2. Let X and Y be two paracompact spaces, ρ : X − Y be a continuous map, U = {U α } α∈A and V = {V β } β∈B be open covers of X and Y respectively, based on arbitrary index sets A and B, and π : A − B be a map such that for any α ∈ A. Let N U and N V be the nerves of U and V, respectively. Observe that π induces the canonical simplicial mapπ : N U − N V since U α 0 ∩ · · · ∩ U αn = ∅ implies V π(α 0 ) ∩ · · · ∩ V π(αn) = ∅, and ρ induces the canonical mapρ : ∆X U − ∆Y V mapping (x, p) to (ρ(x),π(p)).
Then, there exist homotopy equivalences g : X − ∆X U and g : Y − ∆Y V which commute with ρ andρ up to homotopy: is a homotopy equivalence between X and ∆X U , where {ψ α } α∈A is a partition of unity subordinate to the cover U. And, is a homotopy equivalence between Y and ∆Y V where {ψ β } β∈B is a partition of unity subordinate to the cover V.
Hence, one can just construct a homotopy betweenρ • g and g • ρ in the following way: Here, note that the linear interpolation betweenπ (ψ α (x)) α∈A and ψ β (ρ(x)) β∈B is well-defined since, because of the properties of partition of unity and the assumption that ρ(U α ) ⊆ V π(α) , forms a simplex in N V.
Finally, one can prove the functorial nerve lemma.
Proof of Theorem 6. Combine Lemma A.1 and Lemma A.2.
A.4 Proof of VR r (S n ) S n for r ∈ 0, arccos − 1 n+1 In this appendix we will prove Theorem 10.
Theorem 10. For any n ∈ Z >0 , we have VR r (S n ) S n for any r ∈ 0, arccos −1 n+1 . Since the case of S 1 is already proved in [3], it is enough to prove the above theorem for S n with n ≥ 2. Moreover, unlike the other parts of the paper, in this section we discriminate between the simplicial complex VR r (S n ) and its realization |VR r (S n )|.
To prove Theorem 10, we will basically emulate the proof strategy which Hausmann in [46]. However, a crucual modification will be necessary which requires the following version of Jung's theorem. The version of Jung's theorem stated above is different from the one considered by Katz [50,Lemma 2] in the following two senses: (1) We provide a precise formula for the radius ψ(D) of the closed ball covering A, depending on D = diam(A). In particular, our version is stronger when D is small. (2) On the contrary, if D is large (close to arccos − 1 n+1 ), then the radius ψ(D) can be as large as π 2 . But π 2 is strictly greater number than π − arccos − 1 n+1 which is the radius guaranteed by Katz's version. So, for the case when D is large, Katz's version is stronger.
The proof of our version is somewhat similar to the classical proof in [34].
Hence, f is non-increasing, so this concludes the proof.
The following Lemma is an analogue of [46, (3
Proof. Let σ = {x 0 , . . . , x q } be a simplex of VR r (S n ) and let I σ be the image of T σ . If σ is a face of σ then I σ ⊆ I σ , and thus VR δ (I σ ) is a subcomplex of VR δ (I σ ) for all δ > 0. On the other hand, VR δ (I σ ) is acyclic for all δ > 0. Indeed, by Theorem 13, ∃ u ∈ I σ such that I σ ⊂ B π 2 (u, S n ). So, one can consider the obvious crushing from I σ to {x} via the shortest geoedesics. So, VR δ (I σ ) must be contractible by Lemma A.3 and [46, (2.3) Corollary]. These considerations show that for 0 < δ ≤ δ ≤ arccos − 1 n+1 , the correspondence is an acyclic carrier Φ δ,δ from VR δ (S n ) to VR δ (S n ) (see [63, §13]). We now use the theorem of acyclic carrier ( [63,Theorem 13.3]). This implies that there exists an augmentation preserving chain map ν : C * (VR r (S n )) − C * (VR r (S n )) which is carried by Φ r,r . Let µ denote the canonical inclusion from VR r (S n ) into VR r (S n ). Then, φ r ,r is an acyclic carrier for both ν • µ * and the indentity of C * (VR r (S n )). By theorem of acyclic carrier again, these two maps are chain homotopic and thus ν • µ * induces the identity on H * (VR r (S n )). The same argument shows that µ * • ν induces the identity on H * (VR r (S n )) (using the acyclic carrier Φ r,r ).
We will now compare the simplicial homology of VR r (S n ) with the singular homology of M . Formula (9) shows that the correspondence σ − T σ gives rise to a chain map T r : C * (VR r (S n )) − SC * (S n ) where SC * (S n ) denotes the singular chain complex of S n .
Lemma A.5. If 0 < r ≤ arccos − 1 n+1 then the chain map T induces an isomorphism on homology.
Proof. We shall need a few accessory chain complexes. For δ > 0, denote by SC * (S n ; δ) the sub-chain complexes of SC * (S n ) based on singular simplexes τ such that there exists u ∈ S n with the image of τ is contained in the open ball B δ (u, S n ). Recall that the inclusion SC * (S n ; δ) − SC * (S n ) induces an isomorphism on homology ( [46,Theorem 31.5]).
We shall also use the ordered chain complex C * (VR r (S n )): the group C q (VR r (S n )) is free abelian group on (q + 1)-tuples (x 0 , . . . , x q ) such that {x 0 } ∪ · · · ∪ {x q } is a simplex of VR r (S n ). One can view that C * (VR r (S n )) as a sub-chain complex of C * (VR r (S n )) by associating a qsimplex {x 0 , . . . , x q } of VR r (S n ) (with our convention that x 0 < x 1 < · · · < x q for the wellordering on S n ) the (q + 1)-tuple (x 0 . . . , x q ). It is also classical that this inclusion is homology isomorphism ([46, Theorem 3.6]). Observe that the construction σ T σ does not require that the vertices of σ are all distinct. One can then define T σ for a basis element of C * (VR r (S n )) and thus extend to a chain map T r : C * (VR r (S n )) − SC * (S n ; ψ(r)). Now, choose r < r such that ψ(r ) ≤ r 2 . One then has the following commutative diagram: C * (VR r (S n )) SC * (S n ; ψ(r )) C * (VR r (S n )) SC * (S n ; ψ(r)) T r T r Let τ : ∆ q − S n be a singular simplex whose image is contained in some open ball of radius ψ(r ). The (q + 1)-tuple (τ (e 0 ), . . . , τ (e q )) is element of C q (VR r (S n )). This correspondence gives rise to a chain map R : SC * (S n ; ψ(r )) − C * (VR r (S n )) The composition R • T r is equal to the canonical inclusion C * (VR r (S n )) ⊂ C * (VR r (S n )) which induces a homotopy isomorphism by Lemma A.4. Let us now understand the composition T r • R : SC * (S n ; ψ(r )) SC * (S n ; ψ(r)). Let τ : ∆ q − S n be a singular simplex such that τ (∆ q ) ⊂ B ψ(r ) (y, S n ) for some y ∈ S n . Therefore, τ := T r • R(τ ) also satisfies τ (∆ q ) ⊂ B ψ(r ) (y, S n ) since ψ(r ) < π 2 . Hence, τ and τ are canonically homotopic (following, for each s ∈ ∆ q , the shortest geodesic joining τ (s) to τ (s)). As in the proof of the homotopy axiom for singular homology ( [46, §30]), these provide a chain homotopy between T r • R and the inclusion SC * (S n ; ψ(r )) ⊂ SC * (S n ; ψ(r)). As said before, this inclusion is known to induce a homology isomorphism. Therefore, T r • R induces an isomorphism on homology.
We have shown that both R • T r and T r • R induce homology isomorphisms. Therefore, T r induces a morphism both injective and surjective, hence homology isomorphism.
Proof. Let γ : [0, 1] − S n represent an element of π 1 (S n ). Choose large enough positive integer N such that 1 N is smaller than the Lebesgue number for the covering {γ −1 (B r 2 (x, S n ))} x∈S n . Then, d S n γ k N , γ k+1 N < r for any k = 0, . . . , N − 1. Hence the path γ| [( k N ),( k+1 N )] is then canonically homotopic to a parametrization of the shortest geodesic joining γ k N to γ k+1 N .
This shows that γ is homotopic to a composition γ of geodesics in open balls of radius r 2 . Such a path γ represents the image of T of an element of π 1 (|VR r (S n )|), the latter being identified with the edge-path group of the simplicial complex VR r (S n ) ( [71, p.134-139]). This proves that π 1 T : π 1 (|VR r (S n )|) − π 1 (S n ) is surjective. Now, to prove the injectivity, suppose π 1 T ([α]) = 0 where α : [0, 1] − |VR r (S n )| is a continuous map satisfying α(0) = α(1). Moreover, again by [71, p.134-139], one can assume α is induced by an edge-path of VR r (S n ). In other words, there are a positive integer N , and x 0 , . . . , x N −1 , x N = x 0 ∈ S n such that d S n (x i , x i+1 ) < r and α i N = x i for i = 0, . . . , N − 1 (here, we view x i as 0-simplex). Next, by the assumption, We are now in position to state the proof of Theorem 10.
Proof of Theorem 10. As mentioned in the beginning of this section, one can assume n ≥ 2. Hence, S n is simply connected. Therefore, by Lemma A.6, |VR r (S n )| is also simply connected. Also, by Lemma A.5 and the isomorphism between simplicial and singular homology [63, §34], T induces an isomorphism on homology. Therefore, T is homotopy equivalence by [45,Corollary 4.33].

A.5 Simplicial proof of Theorem 8
In this section, we will provide an alternative proof of Theorem 8 which takes place at the level of simplicial complexes.
Lemma A.7. Suppose that a compact metric space (X, d X ), a field F, and a nonnegative integer k are given. Then, for every I ∈ barc VR k (X; F), (ii) if v ∈ [0, ∞) is the right endpoint of I, then v ∈ I (i.e. I is right-closed).
Proof of (i). The fact that I ∈ barc VR k (X; F) implies that, for each r ∈ I, there exists a simplicial k-cycle c r on VR r (X) with coefficients in F satisfying the following: 1. [c r ] ∈ H k (VR r (X); F) is nonzero for any r ∈ I. Now, suppose that u is a closed left endpoint of I (so, u ∈ I). In particular, by the above there exists a simplicial k-cycle c u on VR u (X) with coefficients in F with the above two properties. Then, c u = l i=1 α i σ i where α i ∈ F and σ i is a subset of X with cardinality k + 1 and diam(σ i ) < u for any i = 1, . . . , l. Observe that one can choose small ε > 0 such that diam(σ i ) < u − ε for any i = 1, . . . , l. Therefore, if we define c u−ε := l i=1 α i σ i , it is also a simplicial k-cycle on VR u−ε (X). Also, it satisfies (i u−ε,u ) (c u−ε ) = c u obviously since i u−ε,u is the canonical inclusion.
Moreover, c u−ε cannot be null-homologous. Otherwise, there would exist a simplicial (k + 1)chain d u−ε of VR u−ε (X) with coefficients in F such that ∂ So, we must have [c u−ε ] = 0. But, the existence of such c u−ε contradicts the fact that u is the left endpoint of I. Therefore, one concludes that u cannot be a closed left endpoint, so it must be an open endpoint.
Proof of (ii). Now, suppose that v is an open right endpoint of I (so that v / ∈ I and therefore c v is not defined by the above two conditions). Choose small enough ε > 0 so that v − ε ∈ I, and let Then, c v must be null-homologous.
This means that there exists a simplicial (k + 1)-dimensional chain d v of VR v (X) with coefficients in F such that ∂ (v) k+1 d v = c v . Then, d v = l i=1 α i τ i where α i ∈ F and τ i is a subset of X with cardinality k + 2 and diam(τ i ) < v for any i = 1, . . . , l. Observe that one can choose ε ∈ (0, ε] such that diam(τ i ) < v − ε for any i = 1, . . . , l. Therefore, if we define d v−ε := l i=1 α i τ i , it is also a simplicial (k + 1)-dimensional chain on VR v−ε (X). Also, it satisfies (i v−ε ,v ) (d v−ε ) = d v . Then, again by the naturality of boundary operators, but it contradicts the fact that [c v−ε ] = 0. Therefore, v must be a closed endpoint. Now, Theorem 8 is achieved as a direct result of Lemma A.7.