Homotopical decompositions of simplicial and Vietoris Rips complexes

Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced by a cover of its vertices. We study how the homotopy type of such decompositions approximates the homotopy of the simplicial complex itself. The difference between the simplicial complex and such an approximation is quantitatively measured by means of the so called obstruction complexes. Our general machinery is then specialized to clique complexes, Vietoris-Rips complexes and Vietoris-Rips complexes of metric gluings. For the latter we give metric conditions which allow to recover the first and zero-th homology of the gluing from the respective homologies of the components.


Introduction
Homology is an example of an invariant that is both calculable and geometrically informative. These two features are key reasons why invariants derived from homology are fundamental in Algebraic Topology in general and in Topological Data Analysis (TDA) [8] in particular. Calculability is a consequence of the fact that homology converts homotopy push-outs into Mayer-Vietoris exact sequences. Decomposing a space into a homotopy push-out enables us to extract the homologies of the decomposed space (global information) from the homologies of the spaces in the push-out (local information).
Ability of extracting global information from local is important. What is meant by local information however depends on the input and the description of considered spaces. For example what is often understood as local information in TDA differs from the local information described above (push-out decomposition). In TDA the input is typically a finite metric space. This information is then converted into spacial information and in this article we focus on the so called Vietoris-Rips construction [19] for that purpose. Homologies extracted from this space give rise to invariants of the metric space used in TDA such as persistent homologies [7,12,14,16,17], bar-codes [23], stable ranks [10,22], or persistent landscapes [6]. This conversion process, from metric into spacial information, does not in general transform the gluing of metric spaces [24] into homotopy push-outs and homotopy colimits of simplicial complexes. The aim of this paper is to understand how close such data driven decompositions are to decompositions into homotopy push-outs. Our work was inspired by [4] and [5], and grew out of realisation that analogous statements hold true for arbitrary simplicial complexes and not just Vietoris-Rips complexes.
To get these general statements we use categorical techniques. This enables us to prove stronger results using arguments that for us are more transparent.
The most general input for our investigation is a simplicial complex K and a cover X ∪ Y = K 0 of its set of vertices. In this article we study the map K X ∪ K Y ⊂ K where K X and K Y are subcomplexes of K consisting of all these simplices of K which are subsets of X and Y respectively. The goal is to estimate the homotopy fibers of this inclusion. We do that in terms of obstruction complexes St(σ, X ∩ Y ) := {µ ⊂ X ∩ Y | 0 ≤ |µ| and µ ∪ σ ∈ K} indexed by simplices σ in K (see Definition 4.1). Our main result, Theorem 7.5, states that the homotopy fibers of K X ∪ K Y ⊂ K are in the same cellular class (see Paragraph 2.5, and [9,13,15]) as the obstruction complexes St(σ, X ∩ Y ) for all σ in K such that σ ∩ X = ∅, σ ∩ Y = ∅, and σ ∩ X ∩ Y = ∅. For instance (see Corollary 7.6.1) if, for all such σ, the obstruction complex St(σ, X ∩ Y ) is contractible, then K X ∪ K Y ⊂ K is a weak equivalence and consequently K decomposes as a homotopy pushout hocolim(K X ←֓ K X∩Y ֒→ K Y ) leading to a Mayer-Vietoris exact sequence. Another instance of our result (see Corollary 7.6.2) states that if these obstruction complexes have trivial homology in degrees not exceeding n, then so do the homotopy fibers of K X ∪ K Y ⊂ K and consequently this map induces an isomorphism on homology in degrees not exceeding n, leading to a partial Mayer-Vietoris exact sequence. Yet another consequence (see Corollary 7.6.3) is that if these obstruction complexes have p-torsion homology for a prime p, then, for any field F of characteristic different than p, the inclusion K X ∪ K Y ⊂ K induces an isomorphism on homology with coefficients in F leading again to a Mayer-Vietoris exact sequence.
In section 9 we specialise our theorem about the cellularity of the homotopy fibers of the inclusion K X ∪ K Y ⊂ K to the case when K is a clique complex and give some conditions that imply the assumptions of the theorem in this case. Obtained results in principle generalize all the statements proven in [4,5] for Vietoris-Rips complexes. The point we would like to make is that these statements are not about Vietoris-Rips complexes but rather about these complexes being clique. In particular triangular inequality of the input metric space is not needed for our statements to hold. We then prove Theorem 11.5 for which it is essential that considered complex is the Vietoris-Rips complex of the metric gluing of pseudometric spaces for which the triangular inequality is satisfied. This theorem gives 2 connectedness of the relevant homotopy fibers and hence can be used to calculate H 1 and H 0 of the gluing in terms of H 1 's and H 0 's of the components and the intersection.

Small categories and simplicial sets
In this section we recall some elements of a convenient language for describing and discussing homotopical properties of small categories. The key role here is played by the nerve construction [20,21] that transforms small categories into simplicial sets. We refer the reader to [11,18] for an overview of how to do homotopy theory on simplicial sets. We consider the standard model structure on the category of simplicial sets where weak equivalences are given by the maps inducing bijections on all the homotopy groups with respect to any choice of a base point.
Here is a list of definitions and characterizations of various homotopical notions for small categories and some statements regarding these notions.
2.1. Let C be a property of simplicial sets, such as being contractible, n-connected, having p-torsion integral reduced homology, or having trivial reduced homology in some degrees. By definition a small category I satisfies C if and only if its nerve N (I) satisfies C.
2.2. Let C be a property of maps of simplicial sets, such as being a weak equivalence, a homology isomorphism, or having n-connected homotopy fibers. By definition, a functor f : I → J between small categories satisfies C if and only if the map of simplicial sets N (f ) : N (I) → N (J) satisfies C. Assume I has a terminal object t. Then there is a unique natural transformation from the identity functor id : I → I to the constant functor t : I → I with value t. The identity functor is therefore homotopic to the constant functor, and consequently I is contractible. By a similar argument, a category with an initial object is also contractible.
2.4. A commutative square of small categories is called a homotopy push-out (pullback) if after applying the nerve construction the obtained commutative square of simplicial sets is a homotopy push-out (pull-back).
2.5. Recall that a collection C of simplicial sets is closed if it contains a nonempty simplicial set and it is closed under weak equivalencies and homotopy colimits indexed by arbitrary small contractible categories [9,Corollary 7.7]. Any closed collection contains all contractible simplicial sets [9,Proposition 4.5]. If a closed collection contains an empty simplicial set, then it contains all simplicial sets.
The following are some examples of collections of simplicial sets that are closed: contractible simplicial sets, n-connected simplicial sets, connected simplicial sets having p-torsion reduced integral homology, simplicial sets having trivial reduced homology with some fixed coefficients up to a given degree, and more generally simplicial sets which are acyclic with respect to some (possibly not ordinary) homology theory.
Let C be a closed collection of simplicial sets and f : I → J be a functor between small categories. We say that homotopy fibers of f satisfy C if the homotopy fibers of N (f ) : N (I) → N (J), over any component in N (J), belong to C.
2.6. Let f : I → J be a functor between small categories. For an object j in J, the symbol j ↑ f denotes the category whose objects are pairs (i, α : j → f (i)) consisting of an object i in I and a morphism α : j → f (i) in J. The set of morphisms in j ↑ f between (i, α : j → f (i)) and (i ′ , α ′ : j → f (i ′ )) is by definition the set of morphisms β : i → i ′ in I for which the following triangle commutes: The composition in j ↑ f is given by the composition in I.
For an object j in J, the symbol f ↓ j denotes the category whose objects are pairs (i, α : f (i) → j) consisting of an object i in I and a morphism α : is by definition the set of morphisms β : i → i ′ in I for which the following triangle commutes: The composition in f ↓ j is given by the composition in I.
the homotopy fibers of f are n-connected. Thus in this case f induces an isomorphism on homotopy groups in degrees 0, . . . , n and a surjection in degree n + 1. (3) If, for every j, f ↓ j (respectively j ↑ f ) is connected and has p-torsion reduced integral homology in degrees not exceeding n (n ≥ 0), then the homotopy fibers of f are connected and have p-torsion reduced integral homology in degrees not exceeding n. Thus in this case, for primes q = p, f induces an isomorphism on H * (−, Z/q) for * ≤ n and a surjection on H n+1 (−, Z/q). (4) If, for every j, f ↓ j (respectively j ↑ f ) is acyclic with respect to some homology theory, then f is this homology isomorphism.

3.
Simplicial complexes and small categories 3.1. Fix a set U called a universe. A simplicial complex is a collection K of finite nonempty subsets of U that satisfies the following requirement: if σ ⊂ U is in K, then every non-empty subset of σ is also in K. Let X ⊂ U be a subset. The collection {{x} | x ∈ X}, consisting of singletons in X, is a simplicial complex denoted also by X, called the discrete simplicial complex on X. The collection {σ ⊂ X | 1 ≤ |σ| < ∞} of all finite nonempty subsets of X is also a simplicial complex denoted by ∆[X] and called the simplex on X. A simplicial complex is called a standard simplex if it is of the form ∆[X] for some X ⊂ U. The simplex ∆[∅] is called the empty simplex or the empty simplicial complex.
3.2. Let K be a simplicial complex. An element σ in K is called a simplex of K of dimension |σ| − 1. The set of n-dimensional simplices in K is denoted by K n . An element x ∈ U is called a vertex of K if {x} is a simplex in K. The assignment x → {x} is a bijection between the set of vertices in K and the set of its 0-dimensional simplices K 0 . We use this bijection to identify these sets. Thus we are going to refer to 0-dimensional simplices in K also as vertices.
3.3. If {K i } i∈I is a family of simplicial complexes, then both the intersection ∩ i∈I K i and the union ∪ i∈I K i are also simplicial complexes. If K is a simplicial complex and X ⊂ U is a subset, then the intersection K ∩ ∆[X] is a simplicial complex consisting of the elements of K that are subsets of X. This intersection is called the restriction of K to X and is denoted by Thus the intersection of standard simplices (see 3.1) is again a standard simplex, which can possibly be empty.
Let L and K be simplicial complexes. If L ⊂ K, then L is called a subcomplex of K. Being a subcomplex is a partial order relation on the collection of all simplicial complexes which gives this collection the structure of a lattice. The union is the join and the intersection is the meet.
The collection ∪ 0≤i≤n K i is a subcomplex of K called the n-th skeleton of K and denoted by sk n K.

3.4.
A map between two simplicial complexes K and L is by definition a function φ : K → L for which there exists a function f : Thus f is uniquely determined by φ and we often use the symbol φ 0 to denote f . If K and L are fixed, then φ is determined by f = φ 0 . The inclusion L ⊂ K of a subcomplex is an example of a map.
For any simplicial complex K, the inclusions K 0 ⊂ K ⊂ ∆[K 0 ], between the discrete simplicial complex K 0 , K, and the simplex ∆[K 0 ] on K 0 are maps of simplicial complexes. The induced functions on the set of vertices for these two inclusions are given by the identity function id : K 0 → K 0 .
3.5. Classically, the geometrical realization is used to define and study homotopical properties of simplicial complexes. For example, a commutative square of simplicial complexes is called a homotopy push-out (pull-back) if after applying the realization, the obtained commutative square of spaces is a homotopy push-out (pull-back). For instance two simplicial complexes K and L fit into the following commutative diagram of subcomplex inclusions: By applying the realization construction to this square, we obtain a commutative square of spaces which is a push-out and hence a homotopy push-out as the maps involved are cofibrations.
Since the realization of a simplicial complex K can be built from the realization of its n-skeleton sk n K by attaching (possibly in many steps) cells of dimension strictly bigger than n, we get: 3.6. Proposition. Let n ≥ 0 be a natural number. For every simplicial complex L such that sk n+1 K ⊂ L ⊂ K, the homotopy fibers of the inclusion L ⊂ K are n-connected. In particular, the map L ⊂ K induces an isomorphism on homotopy and integral homology groups in degrees 0, . . . , n and a surjection in degree n + 1.
There are situations however when another way of extracting homotopical properties of simplicial complexes is more convenient. In the rest of this section, we recall how one can retrieve and study such information by first transforming simplicial complexes into small categories and then using the nerve construction as explained in Section 2.
3.7. Let K be a simplicial complex. The simplex category of K, denoted also by the same symbol K, is by definition the inclusion poset of its simplices. Thus, the objects of K are the simplices in K and the sets of morphisms are either empty or contain only one element: If φ : K → L is a map of simplicial complexes, then the assignment σ → φ(σ) is a functor of simplex categories. We denote this functor also by the symbol φ : K → L. Not all functors between K and L are of such a form.
The geometrical realization of a simplicial complex is weakly equivalent to the realization of the nerve of this simplicial complex. Thus to describe homotopical properties of simplicial complexes we can either use their geometrical realizations or the nerves of their simplex categories.
3.8. Let K be a simplicial complex and σ be its simplex. Define the star of σ to be St(σ) : The star of any simplex is contractible. More generally: Proof. For all µ in L, the inclusions µ ֒→ µ ∪ (σ \ S) ←֓ σ \ S form natural transformations between: • the identity functor id : L → L, µ → µ, • the constant functor L → L, µ → σ \ S, • and L → L, given by µ → µ ∪ (σ \ S). The identity functor id : L → L is therefore homotopic to the constant functor and consequently L is contractible.
i.e., if for any simplex σ in K, the set σ ∪ τ is also a simplex in K. For example, if X ⊂ U is non empty, then all simplices in ∆[X] (see 3.1) are central. If τ is a central simplex in K, then so is any subset τ ′ ⊂ τ . According to Proposition 3.9, a simplicial complex that has a central simplex is contractible.
3.11. Let K and L be simplicial complexes. If K ∩ L = ∅, then we define their join K * L to be the simplicial complex consisting of all subsets of U of the form σ ∪ τ where σ is in K and τ is in L. The join is only defined for disjoint simplicial complexes. The set of vertices (K * L) 0 is given by the Thus, for any non-empty subset X ⊂ U \ K 0 , the join K * ∆[X] is contractible. Furthermore the join commutes with unions and intersections: if (K 1 ∪K 2 )∩L = ∅, then (K 1 ∪K 2 ) * L = (K 1 * L)∪(K 2 * L) and (K 1 ∩K 2 ) * L = (K 1 * L)∩(K 2 * L). This can be used to show that, for any choice of a base-points in K and L, the join K * L has the homotopy type of the suspension of the smash Σ(|K| ∧ |L]). In particular if K is n-connected and L is m-connected, then K * L is n + m + 1-connected.

One outside point
In this section we recall how the homotopy type of a simplical complex changes when a vertex is added. We start with defining subcomplexes that play an important role in describing such changes. These complexes are essentially used throughout the entire paper. 4.1. Definition. Let K be a simplicial complex and A ⊂ U be a subset. For a simplex σ in K, define the obstruction complex: If µ belongs to St(σ, A), then so does any of its non empty finite subsets. Thus Fix a vertex v in K. Any simplex in K either contains v or it does not. This means K = K K0\{v} ∪ St(v) and hence we have a homotopy push-out square: gives contractibility of St(v). The simplicial complex K fits therefore into the following homotopy cofiber sequence: Here are some basic consequences of this fact:

Corollary.
Let v be a vertex in a simplical complex K.
is n-connected for a natural number n ≥ 0, then the map K K0\{v} ⊂ K induces an isomorphism on homotopy groups in degrees 0, . . . , n and a surjection in degree n + 1.
is connected and has p-torsion reduced integral homology in degrees not exceeding n (n ≥ 0), then for a prime q not dividing p, is acyclic with respect to some homology theory, then K K0\{v} ⊂ K is this homology isomorphism.

Two outside points
Let us fix two distinct vertices v 0 and v 1 in a simplicial complex K. Note that There are two possibilities. First, {v 0 , v 1 } is not a simplex in K. In this case Assume {v 0 , v 1 } is a simplex in K. Then: and thus according to Proposition 3.9 this complex is contractible; ΣSt → K K0\{v0} ∪ K K0\{v1} ֒→ K 6. n + 1 outside points Homotopy cofiber sequences described in Sections 4 and 5 are particular cases of a more general statement regarding an arbitrary number of outside points. The aim of this section is to present this generalization.
Let us fix a set σ = {v 0 , v 1 , . . . , v n } ⊂ K 0 of n + 1 distinct vertices in a simplcial complex K which may not necessarily be a simplex in K. Note that v∈σ (K 0 \ {v}) = K 0 . In this section we are going to investigate the inclusion v∈σ K K0\{v} ⊂ K There are two possibilities. First, σ is not a simplex in K. In this case v∈σ K K0\{v} = K.
Assume σ is a simplex in K. Then: Since the star complex St(σ) is contractible (see Proposition 3.9), K is therefore weakly equivalent to the homotopy cofiber of the map: Next we identify the homotopy type of v∈σ K K0\{v} ∩ St(σ): 6.1. Proposition. Let σ be a simplex of dimension n in a simplicial complex K.
has the homotopy type of the n-th suspension of the obstruction complex Σ n St(σ, K 0 \ σ).
Proof. Consider the inclusion poset of all subsets τ ⊂ σ. For any such subset τ ⊂ σ, define: , we obtain a contra-variant functor indexed by the inclusion poset of all subsets of σ. For example in the case σ = {v 0 , v 1 , v 2 }, this contra-variant functor describes a commutative cube: For arbitrary n, the functor F describes a commutative cube of dimension n + 1. This cube is both co-cartesian and strongly cartesian ([]). It is therefore also a homotopy co-cartesian. By intersecting with St(σ), we obtain a new cube τ → F (τ ) ∩ St(σ). The properties of being co-cartesian and strongly cartesian are preserved by taking such interection. Consequently v∈σ K K0\{v} ∩St(σ) has the homotopy type of hocolim ∅ =τ ⊂σ K K0\τ ∩ St(σ) .
For any proper subset ∅ = τ σ, we have an equality: We can then use Proposition 3.9 to conclude that Thus all the spaces in the cube τ → F (τ ) ∩ St(σ), except for the initial and the terminal, are contractible. That implies that the terminal space We finish this section with summarising the consequences of the discussion leading to Proposition 6.1 and the proposition itself: Then there is a homotopy cofiber sequence: 7. Push-out decompositions I.
In this section our starting assumption is: 7.1 (Starting input I). K is a simplicial complex, X ∪ Y = K 0 is a cover of its set of vertices, and A := X ∩ Y .
By restricting K to X and Y , and taking the union of these restrictions we obtain a subcomplex K X ∪ K Y ⊂ K. Since K X ∩ K Y = K A , this subcomplex fits into the following homotopy push-out square: This push-out can be then used to extract various homotopical properties of the union K X ∪ K Y from the properties of K X , K Y and K A . For example, if K X , K Y and K A belong to a closed collection (see 2.5), then so does K X ∪ K Y . If K A is contractible, then K X ∪ K Y has the homotopy type of the wedge of K X and K Y , and its reduced homology is the sum of the reduced homologies of K X and K Y . More generally, there is a Mayer-Vietoris sequence connecting homologies of A fundamental question discussed in this article is: under what circumstances the inclusion K X ∪ K Y ⊂ K is a weak equivalence, or homology isomorphism, or has highly connected homotopy fibers etc? Such circumstances would enable us to express various homotopical properties of K in terms of the properties of its restrictions K X , K Y and K A .

7.2.
Definition. Under the starting assumption 7.1, define P to be the subposet of K given by: We are going to be more interested in the set of simplices of K that do not belong to P , which explicitly can be described as: The poset P may not be the simplex category of any simplicial complex. There are two poset inclusions that we denote by f and g: Our first general observation is: Proof. We are going to show that, for every σ in P , f ↓ σ is contractible. First assume σ ⊂ X or σ ⊂ Y . Then the object (σ, id : σ → σ) is terminal in f ↓ σ and consequently this category is contractible.
Assume σ ∩ A = ∅. Then, for any object (τ, τ ⊂ σ) in f ↓ σ, the subsets τ , τ ∪ (σ ∩ A), and σ ∩ A of σ are simplices that belong to K X ∪ K Y . We can then form the following commutative diagram in P where the top horizontal arrows represent morphisms in K X ∪ K Y : These horizontal morphisms form natural transformations between: • the identity functor id : . The identity functor id : f ↓ σ → f ↓ σ is therefore homotopic to the constant functor. This can happen only if f ↓ σ is a contractible category.
According to Proposition 7.3, the homotopy fibers of g : P ⊂ K and the inclusion K X ∪ K Y ⊂ K are weakly equivalent. To understand these homotopy fibers, we are going to focus on the categories σ ↑ g and then utilise Corollary 2.8. The functor σ → σ ↑ g fits into the following diagram of natural transformations between functors indexed by K op with small categories as values: where: • ψ σ : St(σ, A) → σ ↑ g assigns to µ in St(σ, A) the object in σ ↑ g given by the pair ψ σ (µ) := (µ ∪ σ, σ ⊂ µ ∪ σ). • φ σ : σ ↑ g → St(σ) assigns to (τ, σ ⊂ τ ) the simplex τ in St(σ). These natural transformations satisfy the following properties: 7.4. Proposition. Let σ be a simplex in K.
(1) If σ is in P , then σ ↑ g is contractible and φ σ : Proof. If σ is in P , then (σ, id : σ → σ) is an initial object in σ ↑ g and hence this category is contractible. That proves (1).
The functor ψ σ : St(σ, A) → σ ↑ g has therefore a homotopy inverse and hence is a weak equivalence which proves (2).
We use Corollary 2.8 and Proposition 7.4 to obtain our main statement describing properties of the homotopy fibers of the inclusion K X ∪ K Y ⊂ K: 7.5. Theorem. Notation as in 7.1 and Definition 7.2. Let C be a closed collection of simplicial sets (see 2.5). Assume that, for every σ in K \ P , the obstruction complex St(σ, A) (see 4.1) satisfies C. Then the homotopy fibers of the inclusion K X ∪ K Y ⊂ K also satisfy C.
The following are some particular cases of the above theorem specialized to different closed collections of simplicial sets. (1) If, for every σ in K \ P (see 7.2), the simplicial complex St(σ, A) (see 4.1) is contractible, then K X ∪ K Y ⊂ K is a weak equivalence. (2) If, for every σ in K \P , the simplicial complex St(σ, A) is n-connected, then the homotopy fibers of K X ∪ K Y ⊂ K are n-connected and this map induces an isomorphism on homotopy groups in degrees 0, . . . , n and a surjection in degree n + 1.
(3) Let p be a prime number. If, for every σ in K \ P , the simplicial complex St(σ, A) is connected and has p-torsion reduced integral homology in degrees not exceeding n, then the homotopy fibers of K X ∪ K Y ⊂ K are connected and have p-torsion reduced integral homology in degrees not exceeding n. Thus in this case, for prime q = p, K X ∪ K Y ⊂ K induces an isomorphism on H * (−, Z/q) for * ≤ n and a surjection on H n+1 (−, Z/q). (4) If, for every σ in K \ P , the simplicial complex St(σ, A) is acyclic with respect to some homology theory, then K X ∪ K Y ⊂ K is this homology isomorphism.
We remark that Corollary 7.6.1 is a generalization of [5, Theorem 2] to abstract simplicial complexes.
Requirements for obtaining n-connected fibers can be weakened:

Proposition. Notation as in 7.1 and 7.2. Let n be a natural number. If
St(σ, A) is n-connected for every σ in (sk n+1 K) \ P , then the homotopy fibers of Proof. Consider the following poset inclusions: According to Proposition 7.3, f is a weak equivalence. The homotopy fibers of g 2 are n-connected by Proposition 3.6. Thus if the homotopy fibers of g 1 are nconnected, then so are the homotopy fibers of the inclusion K X ∪ K Y ⊂ K. To show that the homotopy fibers of g 1 are n-connected it is enough to show that the categories σ ↑ g 1 are n-connected for every σ in P ∪ sk n+1 K. Proposition 7.4 gives that σ ↑ g 1 is contractible if σ is in P , and is weakly equivalent to St(σ, A) if σ is in sk n+1 K \ P . By the assumption St(σ, A) are therefore n-connected.
Theorem 7.5 states that the homotopy fibers of the inclusion K X ∪ K Y ⊂ K belong to the smallest closed collection containing all the complexes St(σ, A) for σ in K \ P . Recall that if a closed collection contains an empty simplicial set, then it contains all simplicial sets, in which case Theorem 7.5 has no content. Thus St(σ, A) being non empty, for all σ in K \ P , is an absolute minimum requirement for Theorem 7.5 to have any content. In most of our statements that follow, the assumptions we make have much stronger global non emptiness consequences of the form: Here is a consequence of having one of these intersections non-empty: Proof. Let v be a vertex in σ∈K1\P St(σ, A). Observe that sk 1 (K) is a disjoint union of K 1 \ P and sk 1 (K) ∩ P . This can fail for sk n (K) if n > 1. For every τ in sk 1 (K), define: If τ τ ′ in sk 1 (K), then τ is in P and hence τ = φ(τ ) ⊂ φ(τ ′ ). In this way we obtain a functor φ : sk 1 (K) → P . The inclusion τ ⊂ φ(τ ), is a natural transformation between the skeleton inclusion sk 1 (K) ⊂ K and the composition: Thus these two functors from sk 1 (K) to K are homotopic. The statement of the proposition is then a consequence of Proposition 3.6.
Proposition 8.1 does not generalise to n > 0. Non-emptiness of the intersection σ∈(skn+1K)\P St(σ, A) does not imply that the homotopy fibers of K X ∪ K Y ⊂ K are n-connected. For an easy example see 10.4. To guarantee n-connectedness of these homotopy fibers we need additional restrictions. For example in the following corollary the assumptions imply that St(σ, A) does not depend on σ in (sk n+1 K)\P :

Corollary. Notation as in 7.1 and 7.2. Let n be a natural number. Assume that one of the following conditions is satisfied:
(1) There is an n-connected simplicial complex L such that, for every simplex The set A is non empty. Furthermore, for every simplex σ in (sk n+1 K) \ P and every finite subset µ in A, the union σ ∪ µ is a simplex in K. (4) A = {v} and, for every simplex σ in (sk n+1 K) \ P , the union σ ∪ {v} is also a simplex in K. Then the homotopy fibers of K X ∪ K Y ⊂ K are n-connected.
Proof. The corollary under the assumption (1) is a direct consequence of Proposition 7.7. The assumption (2) is a particular case of (1) with L = K A . The assumption (3) is a particular case of (1) with L = ∆[A]. Finally, the assumption (4) is a particular case of (3).
Here is another example of a statement whose assumption, referred to as "one entry point", has a global nonemptiness consequence:

Corollary. Notation as in 7.1 and 7.2. Let n be a natural number. Assume
there is an element v in A with the following property. For every simplex τ in K such that τ ∩ (X \ A) = ∅, τ ∩ (Y \ A) = ∅, and |τ ∩ (K 0 \ A)| ≤ n + 2, the union τ ∪ {v} is also a simplex in K. Then, for every simplex σ in (sk n+1 K) \ P , the element v is a central vertex (see 3.10) in St(σ, A). Furthermore the homotopy fibers of K X ∪ K Y ⊂ K are n-connected.
Proof. Let σ be a simplex in (sk n+1 K) \ P . If µ belongs to St(σ, A) then, by applying the assumption of the corollary to τ = σ ∪ µ, we obtain that σ ∪ µ ∪ {v} is a simplex in K and hence µ ∪ {v} is a simplex in St(σ, A). This means that v is central in St(σ, A) (see 3.10). Consequently St(σ, A) is contractible (see 3.10) and the corollary follows from Proposition 7.7.

Clique complexes
Recall that a simplicial complex K is called clique if it satisfies the following condition: a set σ of size at least 2 is a simplex in K if and only if all the two element subsets of σ are simplices in K. Thus a clique complex is determined by its sets of vertices and edges.
If K is clique, then the complexes St(σ, A) satisfy the following properties: 9.1. Proposition. Notation as in 7.1. Assume K is clique. Then: (2) If τ and σ are simplices in K such that τ ∪ σ is also a simplex in K, then Proof. Let µ be a subset of A such that, for every two element subset τ of µ, the set τ ∪ σ is a simplex in K, i.e., τ is in St(σ, A). Then, since K is clique, µ ∪ σ is also a simplex in K. Consequently µ belongs to St(σ, A) and hence St(σ, A) is clique. That proves (1).
To prove (2), first note that the inclusion St(τ ∪ σ, A) ⊂ St(τ, A) ∩ St(σ, A) holds even without the clique assumption. Let µ belong to both St(τ, A) and St(σ, A). This means that µ ∪ τ and µ ∪ σ are simplices in K. Since every 2 element subset of µ∪τ ∪σ is a subset of either µ∪τ or µ∪σ or τ ∪σ, by the assumption it is an edge in K. By the clique assumption, µ∪τ ∪σ is then also a simplex in K and consequently µ is in St(τ ∪ σ, A). This shows the other inclusion Statements (3) and (4) follow from (2).
Recall that an intersection of standard simplices is again a standard simplex (see 3.3). This observation together with Propositions 7.7 and 9.1 gives: 9.2. Corollary. Notation as in 7.1 and 7.2. Assume K is clique and, for every edge τ in K 1 \ P , the complex St(τ, A) is a standard simplex. If, for all simplices σ in (sk n+1 K) \ P , the complex St(σ, A) is non-empty, then the homotopy fibers of the inclusion K X ∪ K Y ֒→ K are n-connected.
Since clique complexes are determined by their edges, one can wonder if, for such complexes, the conclusions of Corollaries 8.2 and 8.3 would still hold true if their assumptions are verified only for low dimensional simplices. Here is an analogue of Corollary 8.2 for clique complexes.

Proposition. Notation as in 7.1 and 7.2. Let n be a natural number. Assume K is clique and that one of the following conditions is satisfied:
(1) There is an n-connected simplicial complex L such that, for every edge τ in The complex K A is n-connected and, for every edge τ in K 1 \ P and every element v in A, the set τ ∪ {v} is a simplex in K. Then the homotopy fibers of the inclusion K X ∪ K Y ֒→ K are n-connected.
Let τ be an edge in K 1 \ P and µ be a simplex in K A . Assume (2). This assumption implies that any two element subset of τ ∪ µ is a simplex in K. Since K is clique, the set τ ∪ µ is a simplex in K and consequently µ is a simplex in St(τ, A). Thus for any τ in K 1 \ P , there is an inclusion K A ⊂ St(τ, A), and hence K A = St(τ, A) for any such τ . The assumption (2) implies therefore the assumption (1) with L = K A . 9.4. Corollary. Notation as in 7.1 and 7.2. Assume K is clique and that one of the following conditions is satisfied: (1) The set A is non empty. Furthermore, for every edge τ in K 1 \ P and every subset µ in A such that |µ| ≤ 2, the union τ ∪ µ is a simplex in K.
(2) A = {v} and, for every edge τ in K 1 \ P , the set τ ∪ {v} is also a simplex in K.
Then the inclusion K X ∪ K Y ֒→ K is a weak equivalence.
Proof. Assume (1). If K 1 \ P is empty, then so is K \ P , and hence P = K. In this case the corollary follows from Proposition 7.3. Assume K 1 \ P is non-empty. Since any two element subset of A is a simplex in K and K is clique, then all finite non-empty subsets of A belong to K and hence K A = ∆[A]. In this case the assumption (1) is a particular case of the condition (2) in Proposition 9.3 for all n as ∆[A] is contractible. Finally note that the assumption (2) is a particular case of (1).
The following is an analogue of Corollary 8.3 which is also referred to as "one entry point". Then the inclusion K X ∪ K Y ֒→ K is a weak equivalence.
Proof. Assume (1). Let σ be a simplex in K \ P . Choose a cover σ = τ 1 ∪ · · · ∪ τ n where τ i is an edge in K 1 \P for all i. Then according to Proposition 9.1, A). Let w be a vertex in St(σ, A). Then it is also a vertex in St(τ i , A) for all i. By the assumption {v, w} is then a simplex in K. Thus all the 2 element subsets of σ ∪{v, w} are simplices in K and hence {v, w} is a simplex in F (σ, A). As this happens for all vertices w in St(σ, A), since St(σ, A) is clique, for every simplex µ in St(σ, A), the set µ ∪ {v} is also a simplex in St(σ, A). The vertex v is therefore central in St(σ, A) and consequently St(σ, A) is contractible. The proposition under assumption (1) follows then from Corollary 7.6.(1).
Condition (2) is a particular case of (1). Assume (3). Let τ be an edge in K 1 \ P . Condition (3) applied to the simplex τ gives that τ ∪ {v} is a simplex in K and hence v is a vertex in St(τ, A). Let w be a vertex in St(τ, A). Condition (3) applied to the simplex τ ∪ {w} gives that {v, w} ⊂ τ ∪ {v, w} are simplces in K. We can conclude (3) implies (1).
We finish this section with a statement referred to as "two entry points". This has been inspired by [5,Theorem 3], in which the gluing of two metric graphs along a path is considered. While in that case the two entry points are the endpoints of the path the graphs are glued along, in our framework they have to satisfy one of the listed properties. In both cases however these couple of points determine the weak equivalence stated. 9.6. Proposition. Notation as in 7.1 and 7.2. Assume K is clique and there are two elements a X and a Y in A with the following properties: • For every edge τ in K 1 such that |τ ∩ A| = 1 and |τ ∩ (X \ A)| = 1, the set Then, for every σ in K\P , the set {a X , a Y } is a central simplex (see 3.10) in St(σ, A), and the inclusion K X ∪ K Y ⊂ K is a weak equivalence.
Proof. Let σ be a simplex in K \ P . Any vertex v in σ is a vertex of an edge τ ⊂ σ that belongs to K 1 \ P . According to the assumption, the sets {v, a X , a Y } ⊂ τ ∪ {a X , a Y } are simplices in K. This, together with the clique assumption on Let µ be a simplex in St(σ, A). To prove the proposition, we need to show the set µ ∪ {a A , a Y } is a simplex in F (σ, A) or equivalently σ ∪ µ ∪ {a X , a Y } is a simplex in K. Let x be an arbitrary element in σ ∩ X, y an arbitrary element in σ ∩ Y , and v an arbitrary element in µ. The sets {x, v}, {y, v}, and {x, y} are simplices in K. Thus according to the assumptions so are {x, v, a X }, {y, v, a Y }, and {x, y, a X , a Y }. Consequently the two element sets {x, a X }, {v, a X }, {y, a X }, {x, a Y }, {v, a Y }, {y, a Y }, {a X , a Y }, {x, y} are simplices in K. Since all the two element subsets of σ ∪ µ ∪ {a X , a Y } are of such a form and K is clique, σ ∪ µ ∪ {a X , a Y } is a simplex in K.

Vietoris-Rips complexes for distances
Let Z be a subset of the universe U (see 3.1). A function d : Z × Z → [0, ∞] is called a distance if it is symmetric d(x, y) = d(y, x) and reflexive d(x, x) = 0 for all x and y in Z. A pair (Z, d) is called a distance space. A distance space (Z, d) is sometimes denoted simply by Z, if d is understood from the context, or by d, if Z is understood from the context. Let (Z, d) be a distance space. The diameter of a non empty and finite subset σ ⊂ Z is by definition diam(σ) := max{d(x, y) | x, y ∈ σ}.
A subset X ⊂ Z together with the distance function given by the restriction of d to X is called a subspace of (Z, d).
Let (Z, d) be a distance space and r be in [0, ∞). By definition, the Vietoris-Rips complex VR r (Z) consists of these non-empty finite subsets σ ⊂ Z for which diam(σ) ≤ r (explicitly: d(x, y) ≤ r for all x and y in σ). Vietoris-Rips complexes are examples of clique complexes (see Section 9).
Let X be a subspace of (Z, d). Then the Vietoris-Rips complex VR r (X) coincides with the restriction VR r (Z) X (see 3.3).
Our starting assumption in this section is: In the rest of this section we are going to reformulate in terms of the distance d on Z some of the statements given in the previous sections regarding the homotopy properties of the inclusion VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) for various r in [0, ∞).
Here is a direct restatement of Proposition 9.
The assumption (1) of Proposition 10.2 can be restated as: (connectivity condition) for every τ in VR r (Z) 1 \ P , the complex St(τ, A) is n-connected, and (independence condition) for all pairs of edges τ 1 and τ 2 in VR r (Z) 1 \ P , there is an equality St(τ 1 , A) = St(τ 2 , A). What if the independence condition is not satisfied? For example consider a distance space Z = {x 1 , x 2 , a 1 , a 2 , a 3 , a 4 , y} with the distance function depicted by the following diagram, where the dotted lines indicate distance 2 and the continuous lines indicate distance 1:  To assure VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) is a weak equivalence assumption 10.3 is not enough and we need additional requirements. For example the following is an analogue of Corollary 9.4. 10.5. Proposition. Notation as in 10.1. Assume 10.3. Let v be an element in A given by this assumption. In addition assume that one of the following conditions is satisfied: Then the inclusion VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) is a weak equivalence.
Proof. If Assumption 10.3 and the condition 1 hold, then so does the assumption 1 of Proposition 9.5. Furthermore the condition 3 implies 2 and the condition 2 implies 1. Thus this proposition is a consequence of Proposition 7.1.
The intersection condition of the assumption (2) in Proposition 10.2 requires a choice of a parameter r. The following is its universal version where no parameter is required: 10.6 (Assumption II). Notation as in 10.1. The set A is non empty and for every x in X \ A, y in Y \ A, and v in A, the following inequalities hold d(x, y) ≥ d(x, v) and d(x, y) ≥ d(y, v). Assumption 10.6 has an intuitive interpretation in terms of angles when Z is a subspace of the Euclidean space. In such a setting this condition means that every triangle xvy with vertices x in X \ A, y in Y \ A and v in A, the angle at v must be at least 60 • . We therefore refer to this assumption as the 60 • angle condition. Proof. We already know that the proposition holds if r ≥ diam(A). Assume r < diam(A). We claim that in this case VR r (Z) = P (see 7.2). If not, there are x in X \ A and y in Y \ A such that d(x, y) ≤ r. The assumption would then lead to the following contradictory inequalities r ≥ d(x, y) ≥ diam(A) > r. Thus in this case VR r (Z) = P and the proposition follows from Proposition 7.3.

Metric gluings
A distance d on Z is called a pseudometric if it satisfies the triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in Z.
11.1. Notation as in 10.1. Assume that the distance d on Z is a pseudometric. Let x be in X \ A and y be in Y \ A. For all a in A, by the triangular inequality, d(x, y) ≤ d(x, a) + d(a, y), and hence: The pseudometric space (Z, d) is called metric gluing if the above inequality is an equality for all x in X \ A and y in Y \ A.
If A is finite, then the pseudometric (Z, d) is a metric gluing if and only if, for every x in X \ A and y in Y \ A, there is a in A such that d(x, y) = d(x, a) + d(a, y).
If d X is a pseudometric on X and d Y is a pseudometric on Y such that d X (a, b) = d Y (a, b) for all a and b in A, then the following function defines a pseudometric on Z which is a metric gluing: ) is a metric gluing and A is finite, then, for any edge σ = {x, y} in VR r (Z) \ P , there is a in A such that r ≥ d(x, y) = d(x, a) + d(a, y). Thus in this case the obstruction complex St(τ, A) is non-empty as it contains the vertex a. To assure contractibility of St(τ, A) we need additional assumptions, for example: The simplex assumption can be reformulated as follows: for any vertex v in a simplex σ in VR r (Z) \ P , the complex St(v, A) is a standard simplex (see 3.1). Since the intersection of standard simplices is again a standard simplex, under Assumption 11.2, an obstruction complex St(σ, A), for an arbitrary simplex σ in VR r (Z) \ P , is contractible if and only if it is non empty. This, together with the discussion at the end of 11.1 and Corollary 9. The assumptions of Proposition 11.3 are not enough to guarantee the nonemptiness of the obstruction complexes for simplices in VR r (Z) \ P of dimension 2 and higher. For example consider Z = {x 1 , x 2 , a 1 , a 1 , y} with the distance function depicted by the following diagram, where the dotted lines indicate distance 4, the dashed lines indicate distance 3, the squiggly lines indicate distance 2 and the continuous lines indicate distance 1: To assure isomorphism on π 1 , the simplex assumption 11.2 should be strengthened.
11.4 (Strong simplex assumption). Notation as in 10.1 and 7.2. Let r be in [0, ∞). For any vertex v in an edge σ in VR r (Z) \ P , if a and b are elements in A such that d(a, v) ≤ r and d(v, b) ≤ r, then 2d(a, b) ≤ d(a, v) + d(v, b).
Note that the strong simplex assumption 11.4 implies the simplex assumption 11.2. 11.5. Theorem. Notation as in 10.1 and 7.2. Let r be in [0, ∞). Assume A is finite and (Z, d) is a metric gluing that satisfies the strong simplex assumption 11.4. Then, for any simplex σ in VR r (Z) \ P such that either |σ ∩ X| = 1 or |σ ∩ Y | = 1, the obstruction complex St(σ, A) is contractible. The homotopy fibers of the inclusion VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) are simply connected and this map induces an isomorphism on π 0 and π 1 and a surjection on π 2 .
Proof. Since the strong simplex assumption 11.4 is satisfied, then so is the simplex assumption 11.2 and consequently any obstruction complex St(σ, A) is a simplex. Thus St(σ, A) is contractible if and only if it is non empty.
We are going to show by induction on the dimension of a simplex a more general statement: Under the assumption of Theorem 11.5, for every simplex σ in VR r (Z) \ P for which σ ∩ X = {x 1 , . . . , x n } and σ ∩ Y = {y}, if (a 1 , . . . , a n ) is a sequence in A such that d(x i , y) = d(x i , a i ) + d(a i , y) for every i, then there is l for which a l is in St(σ, A) (d(x i , a l ) ≤ r for all i).
If σ = {x, y} is such an edge, then the statement is clear. Let n > 1 and assume that the statement is true for all relevant simplices of dimension smaller than n. Let σ be in VR r (Z)\P be such that σ∩X = {x 1 , . . . , x n } and σ∩Y = {y}. Choose a sequence (a 1 , . . . , a n ) in A such that d(x i , y) = d(x i , a i )+ d(y, a i ) for every i.
By the inductive assumption, for every j = 1, . . . , n, the statement is true for τ j = σ j \ {x j } and the sequence (a 1 , . . . , a j , . . . , a n ) obtained from (a 1 , . . . , a n ) by removing its j-th element. Thus for every j = 1, . . . , n, there is a s(j) such that s(j) = j and d(x i , a s(j) ) ≤ r for all i = j. If, for some j, d(x j , a s(j) ) ≤ r, then a s(j) would be a vertex in St(σ, A), proving the statement. Assume d(x j , a s(j) ) > r for all j. If j = j ′ , then d(x j , a s(j ′ ) ) ≤ r and d(x j , a s(j) ) > r, and hence a s(j) = a s(j ′ ) . It follows that s is a permutation of the set {1, . . . , n}. This together with the strong simplex assumption leads to a contradictory inequality: (d(x i , a i ) + d(y, a i )) ≤ nr Note that, for any simplex σ in sk 2 VR(Z) \ P , either |σ ∩ X| = 1 or |σ ∩ Y | = 1. Thus, for any such simplex, the obstruction complex St(σ, A) is contractible. We can then use Proposition 7.7 to conclude that the homotopy fibers of the inclusion VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) are simply connected.
The conclusion of Theorem 11.5 is sharp. Its assumptions are not enough to assure that the homotopy fibers of the map VR r (X) ∪ VR r (Y ) ֒→ VR r (Z) are 2 connected. We finish this section with an example illustrating this fact. 11.6. Let Z = {x 1 , x 2 , a 11 , a 12 , a 21 , a 22 , y 1 , y 2 }, X = {x 1 , x 2 , a 11 , a 12 , a 21 , a 22 } and Y = {y 1 , y 2 , a 11 , a 12 , a 21 , a 22 }. Consider the distance function d on Z described by the following table: The distance d satisfies the triangular inequality and hence (Z, d) is a metric space. Furthermore d(x i , y j ) = d(x i , a ij )+d(a ij , y j ) for any i and j. Thus (Z, d) is a metric gluing of X and Y . The metric space (Z, d) can be represented by the following diagram, where the continuous lines or no line indicate distance 8 or smaller and the dotted lines indicate distance 9: a 11 y 1 a 21 x 1 a 12 y 2 x 2 a 22 By direct calculation one checks that, for r = 8 and Z = X ∪ Y , the metric space (Z, d) satisfies the strong simplex assumption 11.4. However, the complex VR 8 (X) ∪ VR 8 (Y ) is contractible and VR 8 (Z) is weakly equivaent to S 3 (3-dimensional sphere).The homotopy fiber of VR 8 (X) ∪ VR 8 (Y ) ⊂ VR 8 (Z) is therefore weakly equivalent to the loops space ΩS 3 and hence is not 2 connected.
Here are steps that one might use to see that VR 8 (Z) is weakly equivalent to S 3 . Consider the simplex {x 1 , x 2 } in VR 8 (Z). According to Corollary 6.2, there is a homotopy cofiber sequence: The complexes in this sequence have the following homotopy types: σ ∪µ is a simplex in V R 3 (X). This implies that V R 3 (X) is the join of VR 3 (X ′ ) and VR 3 (X ′′ ) (see Paragraph 3.11) and hence it is weakly equivalent to Σ(S 0 ∧S 1 ) ≃ S 2 .