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Kunneth Theorems for Vietoris-Rips Homology

We prove a Kunneth theorem for the Vietoris-Rips homology and cohomology of a semi-uniform space. We then interpret this result for graphs, where we show that the Kunneth theorem holds for graphs with respect to the strong graph product. We finish by computing the Vietoris-Rips cohomology of the torus endowed with diferent semi-uniform structures.


Introduction
The Vietoris-Rips complex was first defined by Vietoris in 1927 as a way to obtain homology groups from metric spaces [14], and, somewhat later, it began to be used in the study of hyperbolic groups [8]. With the rise of topological data analysis in the last fifteen years, the Vietoris-Rips homology has become a computational, as well as theoretical, tool, and, indeed, it has become the standard invariant used in the homological analysis of data, in addition to its natural importance in the homological analysis of networks and graphs. Despite this surge of popularity, however, relatively little is known about the properties of Vietoris-Rips homology, and, until recently, even many basic results on the Vietoris-Rips homology had not been established. In a previous article [11], the first author introduced a construction of the Vietoris-Rips homology for semi-uniform spaces and proved a variant of the Eilenberg-Steenrod axioms adapted to this context.
Studying the Vietoris-Rips homology and cohomology from the point of view of semi-uniform spaces has many advantages over working directly with clique complexes. The first is that setting up the Vietoris-Rips homology in this way enables one to use variations of classical topological arguments in settings that are very far from topological. Indeed, the results in this paper provide one example of this procedure. This, in turn, leads to another major advantage of working in the category of semi-uniform spaces. As the first author showed in [12], the Vietoris-Rips homology and cohomology are functors which act directly on semi-uniform spaces and uniformly continuous maps between them. Since there exist non-trivial uniformly continuous maps between many different kinds of spaces of interest, in particular, between topological spaces, metric spaces with a preferred scale, and graphs, this functoriality enables the construction of homomorphisms which can be used to compare the resulting Vietoris-Rips homology and cohomology groups across different classes of spaces. This is much more difficult to do if one simply studies the Vietoris-Rips homology in each setting independently of the others. Finally, using the constructions of semi-uniform spaces fromČech closure spaces given in [12], we are able to see that the Vietoris cohomology of Dowker [4], the metric cohomology of Hausmann [9], the Vietoris-Rips cohomology on metric spaces with a preferred scale introduced in [12], and the cohomology of the flag complex of a graph are all instances of the same construction on different semi-uniform spaces.
In this article, we continue the development of the algebraic topology of semiuniform spaces started in [12], first giving an alternate definition of the Vietoris-Rips complex of a relation using simplicial sets, and then studying the Vietoris-Rips homology and cohomology of products of semi-uniform spaces with the goal of establishing Künneth theorems in this context. For the Vietoris-Rips homology as defined here, we will see that, while the Kunneth theorem holds for Vietoris-Rips cohomology on arbitrary semi-uniform spaces, it is not true in general for Vietoris-Rips homology. Nonetheless, we are able to show that it does hold for semi-uniform spaces induced by graphs, which, in turn, implies a Kunneth theorem for the classical Vietoris-Rips homology of graphs, which is the case of most interest to applications. Note that, while it is well-known that the Kunneth theorem is false for the Vietoris-Rips homology using Cartesian products of graphs, by translating the problem into the setting of semi-uniform spaces, we see that that one should use the strong graph product instead. For a homology theory on graphs which satisfies the Kunneth theorem with respect to the Cartesian graph product, see [7].
A Kunneth theorem for the classical Vietoris-Rips homology on metric spaces with respect to the maximum metric on the product may also be deduced by applying the Kunneth theorem for simplicial complexes to the isomorphism in Proposition 10.2 in [1]. However, with the exception of the cases treated in [11], it remains unclear for which cases the the classical and the semi-uniform Vietoris-Rips homology theories coincide. There has also been some recent work on Kunneth theorems in persistent homology [2,6], in which an expression for the persistent homology of a product is obtained, given a filtered complex constructed from a category whose homology has a Kunneth formula. The main contribution of this article is that, by constructing the Vietoris-Rips homology and cohomology in the more general context of semi-uniform spaces, we are able to treat Kunneth theorems for the Vietoris-Rips cohomology of graphs, metric spaces, and even topological spaces as particular instances of the same theorem. Combined with the results of [12], this allows for the computation of the Vietoris-Rips cohomology of products of spaces which are semi-uniformly homotopy equivalent to spaces whose Vietoris-Rips cohomologies are known (such as such as (S 1 × S 1 , d max ), studied in [1]), something which is unattainable using current techniques.

Semi-uniform spaces and the Vietoris-Rips complex
In this section, we recall the definition of semi-uniform spaces, which will be our main object of study. We begin with a few preliminary definitions.
Definition 2.2. Let X be a set, and let F be a non-empty collection of subsets of X with ∅ / ∈ F . We say that F is a filter iff Remark 2.3. The condition that F does not contain the empty set is occasionally additional in the literature, and such filters are sometimes called proper filters. Since we will only be dealing with such filters, we see no need to make such a distinction here. Also, note that the requirement that ∅ / ∈ F combined with Condition 2 of Definition 2.2 above implies that the intersection of any finite collection of sets in F is nonempty.
Definition 2.4. Let X be a set. We say that a filter U on the product X × X is a semi-uniform structure on X iff (1) Each element of U contains the diagonal, i.e. ∆ ⊂ U for all U ∈ U (2) If U ∈ U, then U −1 contains an element of U The pair (X, U), consisting of a set X and a semi-uniform structure U on X, is called a semi-uniform space.
Remark 2.5. Note that, since U is a filter, Condition 2 in Definition 2.4 is equivalent to the condition that U −1 ∈ U.
We give several important examples of semi-uniform spaces.
Example 2.6. Let G = (V, E) be an undirected graph. We denote by σ(G) : Since G is undirected in the above definition, E = E −1 , it follows from Theorem 23.A.4 in [3], that σ(G) is a semi-uniform space.
Example 2.7. Let (X, d) be a metric space. For every q > 0, we define Now fix an r > 0, and define U r to be the semi-uniform structure generated by the sets U r+ǫ for all ǫ > 0, and define U ≤r to be the semi-uniform structure generated by the single relation U ≤r . These are semi-uniform structures by Theorem 23.A.4 in [3].
Example 2.8 (See also [12]). Let X be a set. We define aČech closure operator c : P(X) → P(C) to be a map such that The pair (X, c) will be called aČech closure space. We define the interior of a set A ⊂ X to be For every interior cover U of aČech closure space (X, c), we define two relations V U and R U by Let I c denote the collection of interior covers of (X, c), and let V c and R c be the filters on X × X generated by the collections {V U } U ∈Ic and {R U } U ∈Ic , respectively. As shown in [12], V c and R c are semi-uniform structures on the set X.
In particular, this construction produces a semi-uniform space for each topological space (X, c τ ), where c τ is a closure operator with c 2 τ = c τ . Note that in this case V cτ = R cτ , since both are generated by open covers.
The above examples show how semi-uniform spaces are a simultaneous generalization of graphs, metric spaces with a preferred scale, and closure spaces. We refer to [3,12] for further details on semi-uniform spaces.

The Vietoris-Rips Homology and Cohomology of a Semi-Uniform
Space. Let (X, U) be a semi-uniform space and U ∈ U. We define a simplicial set X U by setting X U 0 := {x ∈ X} = X, and, for n ∈ N, we define . , x n ). We emphasize that the x i need not be distinct points of X, and, in particular, since the diagonal ∆ is a member of every U ∈ U, the X U n will contain elements with Definition 2.9. We call the simplicial set X U the Vietoris-Rips complex of the pair (X, U ).
For each n ∈ Z, let C n (X U ) be the graded free abelian group generated by the elements of the sets X U n , and let C * (X U ) = ⊕ n∈Z C n (X U ). Define a differential d n : is now a chain complex, and we denote its homology by H * (X U ).
Let G be an R-module for a commutative ring R. Then (C * (X U ) ⊗ Z G, d ⊗ 1) is a chain complex over R, and we denote its homology, the homology of X U with coefficients in G, by H * (X U ; G). Similarly, a q-dimensional cochain f ∈ C q (X U ; G) is defined as a homomorphism f : C q (X U ) ⊗ Z R → G and the coboundary is given by for each (q + 1)-simplex σ of X U . This leads to the cohomology groups H * (X U ; G).
We now consider maps between the simplicial sets corresponding to different elements of a semi-uniform structure U.
Since this is both a simplicial map and an inclusion, the final statement follows, and the proof is complete.
We define a partial order ≤ on the semi-uniform structure U by writing U ≤ V iff V ⊆ U . Furthermore, since U is a filter, for any U, V ∈ U, W = U ∩ V ∈ U, and therefore U with this partial order is a directed set. The induced maps φ UV * : into inverse and direct systems of abelian groups, respectively. We finally define the Vietoris-Rips homology and cohomology of a semi-uniform space (X, U) to be We will typically suppress the semi-uniform structure U when it is unambiguous.
Remark 2.11. Note that, since the ordered and unordered simplicial chain complexes are chain equivalent (by [13], Theorem 4.3.8), and since degenerate simplices in the simplicial set do not contribute to the resulting homology and cohomology (by [10], VIII.6), the Vietoris-Rips homology and cohomology defined here for a semi-uniform space (X, U) are isomorphic to the those defined in [11]. Example 2.13. Let (X, d) be a metric space, and let (X, U r ) be the semi-uniform spaces constructed in Example 2.7. Since the sets U ǫ , ǫ > 0 are cofinal in U 0 , it follows that the Vietoris-Rips cohomology of (X, U 0 ) is isomorphic to the metric cohomology studied by Hausmann in [9].
Example 2.14. Let (X, c τ ) be a topological closure space (where c 2 τ = c τ ), and let (X, V cτ ) be the corresponding semi-uniform space from Example 2.8. By results in Dowker [4], the Vietoris-Rips homology and cohomology of (X, V cτ ) are isomorphic to theČech homology and cohomology of the topological space (X, τ ), where the topology τ is generated by

Products
In this section, we recall the definitions of products for semi-uniform spaces and simplicial sets, and we then prove a theorem relating the products of Vietoris-Rips complexes which will be the basis for the Kunneth Theorems to follow.

Products of Semi-Uniform Spaces.
Definition 3.1. Let {(X a , U a )} a∈A be a family of semi-uniform spaces indexed by A. Let X := Π a∈A X a be the Cartesian product of the sets, and we denote by U the filter generated by the subsets of X × X of the form where F ⊂ A is some finite subset of A, U a ∈ U a for each a ∈ F , and π a : X → X a is the projection to the a-th coordinate.
Proposition 3.2. (X, U) defined as above is a semi-uniform space.
Proof. First, let (x, x) ∈ ∆ X×X . Since the ∆ Ua ⊂ U a for every U a ∈ U a and every a ∈ A, it follows that (x, x) is in every element of U. Since (x, x) ∈ ∆ X×X was arbitrary, ∆ X×X is contained in every element ∈ U. Suppose now that U ∈ U. Then U contains a set V of form (1) above. Since, for any U a ∈ U a we have that U −1 a ∈ U a , we see from V −1 a ⊂ U −1 a that V −1 is of the form (1) as well. However, V −1 ⊂ U −1 , so U −1 ∈ U, and the proof is complete. Definition 3.3. We call (X, U) the semi-uniform product of {(X a , U a )} a∈A . We will sometimes denote (X, U) as (Π a∈A X a , Π a∈A U a ).

Products of Simplicial Sets.
Definition 3.4. Let Σ and Σ ′ be simplicial sets. Then the product Σ × Σ ′ is given by (Σ × Σ ′ ) n := {(σ, σ ′ ) | σ ∈ Σ n , σ ′ ∈ Σ ′ n } For any simplicial sets Σ and Σ ′ , the Eilenberg-Zilber theorem [5] gives a quasiisomorphism between the chain complexes C * (Σ×Σ ′ ) and C * (Σ)⊗C * (Σ ′ ). In order to establish the Kunneth theorems for Vietoris-Rips homology, we must further establish a relationship between the Vietoris-Rips complexes of products of semiuniform spaces on the one hand, and the products of Vietoris-Rips complexes of semi-uniform spaces on the other. This follows easily from the respective definitions, and is accomplished in the following theorem. First, however, we make the following remark.
Remark. Let (X, U) and (Y, V) be semi-uniform spaces and let U ∈ U and V ∈ V. Observe that the set U × V ⊂ (X × X) × (Y × Y ) can be seen as a subset of (X × Y ) × (X × Y ) via the isomorphism Note that the expression (X ×Y ) U×V is an abuse of notation, and is, more precisely, (X ×Y ) ψ −1 (U×V ) . Since U ×V ∼ = ψ −1 (U ×V ), however, we will use the first notation in place of the second throughout.
Theorem 3.5. Let (X, U) and (Y, V) be semi-uniform spaces. Then, for any U ∈ U and V ∈ V, we have By definition, ((x 0 , y 0 ), . . . , (x k , y k )) ∈ (X × Y ) U×V iff, for all i, j ∈ {1, . . . , k}, one of the following holds: x j ) ∈ U and y i = y j This, in turn, is true iff (x 0 , . . . , x k ) ∈ X U and (y 0 , . . . , y k ) ∈ Y V . Therefore, φ gives an isomorphism between (X × Y ) U×V and X U × Y V , and the proof is complete.

The Kunneth Theorems
The Kunneth theorems for the Vietoris-Rips homology now follow from the above product relations, the Kunneth theorems for simplicial sets, and the properties of exact sequences in direct and inverse limits. We begin with the results for Vietoris-Rips cohomology, where we have a Kunneth formula in general.
In order to establish this, we will first require the following lemma. We begin by recalling the following definition.
Definition 4.1. We say that a graded module {C q } is said to be of finite type if it is finitely generated for every q.  Let G and G ′ be modules over a principal ideal domain R such that G * G ′ = 0. If H * (X U ; R) and H * (Y V ; R) are of finite type or if H * (Y V ; R) is of finite type and G ′ is finitely generated, then for any q ∈ Z there is a natural short-exact sequence Furthermore, this sequence splits, but not canonically.
We now give the Kunneth theorem for Vietoris-Rips cohomology.
Theorem 4.4. Let (X, U) and (Y, V) be semi-uniform spaces, and let G and G ′ be modules over a principal ideal domain R such that G * G ′ = 0. Suppose that U and V are generated by collections {U α } α∈A and {V β } β∈B such that H * (X Uα ; R) and H * (Y V β ; R) are of finite type for all α ∈ A and β ∈ B, or such that H * (Y V β ; R) is of finite type for β ∈ B and G ′ is finitely generated. Then for any q ∈ Z there is a natural short-exact sequence Furthermore, this sequence splits, but not canonically.
Proof. We first note that the family of sets in U × V of the form U × V are cofinal in U × V. The theorem now follows from Proposition 4.3, the fact that T or 1 and ⊗ commute with direct limits, and that the direct limit is an exact functor.
For homology, the Kunneth theorem does not hold in general, due to the failure of exactness of the inverse limits of exact sequences. Nonetheless, in some special cases, the exact sequences related to elements U × V ∈ U × V prove to be useful. We give two such situations below, beginning, as above with the following lemma.
Lemma 4.5. Let (X, U) and (Y, V) be semi-uniform spaces, and let G and G ′ be modules or a principal ideal domain such that G * G ′ = 0. Then for any U ∈ U, V ∈ V, and q ∈ Z, there is a natural short-exact sequence Furthermore, this sequence splits, but not canonically.
Theorem 4.6. Let (X, U) and (Y, V) be semi-uniform spaces, and let G and G ′ be modules over a principal ideal domain such that G * G ′ = 0. Suppose U * and V * are maximal in U and V, respectively, ordered by inclusion. Then U * × V * is maximal in U × V, and for all q ∈ Z, there exists a natural short-exact sequence Furthermore, this sequence splits, but not canonically.
Proof. We first show that U * × V * is maximal in U × V. Suppose that there exists a W ∈ U × V with W U * × V * . Then there exist U ∈ U and V ∈ V such that U × V ⊂ W U * × V * , a contradiction. Therefore U * × V * is maximal in U × V. For any semi-uniform space (X, U), we have, by definition, If U * is maximal in U, then H * (X U * ; G) is cofinal in the inverse system {H * (X U ; G), φ UV * , U}, from which it follows that as desired. Putting these together, the result now follows from Theorem 4.5.
Although the short exact sequence doesn't hold in general for Vietoris-Rips homology, if the torsion term vanishes on a cofinal subset of the bases for the product semi-uniform structure, we may still conclude that there is an isomorphism of the respective homology groups, as we see from the following.
Theorem 4.7. Let (X, U) and (Y, V) be semi-uniform spaces, and let G and G ′ be modules over a principal ideal domain such that G * G ′ = 0. Suppose that F ′ ⊂ U ×V is a cofinal collection of sets in of the form U × V , U ∈ U, V ∈ V such that, for

Kunneth Theorems for Vietoris-Rips homology on graphs
In this section, we apply the general Kunneth formulae on semi-uniform spaces from Section 4 to prove the Kunneth Theorem for the Vietoris-Rips homology on graphs. We begin with the following construction. All graphs are undirected.
One of the following holds: Proof. By construction, E G has a maximal element E consisting of the edges of the graph G and the diagonal ∆ V , from which it follows that is exactly the homology of the simplicial set generated by the clique complex Σ G of G. Therefore, , and the proof is complete.
In order to prove the Kunneth Theorem for undirected graphs, we will need the following Proposition 5.3. Let G = (V, E) and G ′ = (V ′ , E ′ ) be graphs. Then where the product on the right is the product of semi-uniform spaces.
Proof. We must show that E G × E G ′ = E G⊠G ′ . First, let U ∈ E G × E G ′ . Then, by construction, there exist sets e ∈ E G and e ′ ∈ E G ′ such that e × e ′ ⊂ U . By definition of E G and E G ′ , however, E ⊂ e and E ′ ⊂ e ′ , so E × E ′ ⊂ e × e ′ ⊂ U . Since E × E ′ ∈ E G⊠G ′ and E G⊠G ′ is a filter, therefore U ∈ E G⊠G ′ , and we see that Now suppose U ∈ E G⊠G ′ . Then E × E ′ ⊂ U , and therefore U ∈ E G × E G ′ as well, by definition of the product semi-uniform structure. Therefore Proof. The theorem follows immediately from Proposition 5.3, Theorem 4.6, and the definition of the classical Vietoris-Rips complex for graphs.

Applications to Metric Spaces
As an illustration of the above results, we examine the Vietoris-Rips homology and cohomology of the torus with different semi-uniform structures. We first recall the definition of the classical Vietoris-Rips complex on a metric space and the corresponding relations which can be used to generate semi-uniform structures. Definition 6.1. Suppose that (X, d) is a metric space and r > 0 is a real number. The Vietoris-Rips complex V R < (X; r) is the simplicial complex with vertex set X, where a finite subset σ ⊆ X is a simplex if only if the diam(σ) < r. The Vietoris-Rips complex V R ≤ (X; r) is the simplicial complex with vertex set X, where a finite subset σ ⊆ X is a simplex if only if the diam(σ) ≤ r.
Remark 6.2. For a given r > 0, note that V R < (X; r) is the geometric realization of the simplicial set X Ur , and V R ≤ (X; r) is the geometric realization of the simplicial set X U ≤r . They therefore have the same homology and cohomology groups.
We now recall following theorem from [1]. Theorem 7.4). Denote the circle with unit circumference by S 1 , and consider S 1 as a metric space with the geodesic distance. For 0 < r < 1 2 , suppose that 1 2l+1 < r < l+1 2l+3 for some l ∈ {0, 1, ...}. Then we have a homotopy equivalence The following computation now follows from Theorem 6.3 and Theorem 4.4.
If l = l ′ , then otherwise.
Since the torsion term in the above exact sequence is trivial, the result follows.