Kunneth Theorems for Vietoris-Rips Homology
aa r X i v : . [ m a t h . A T ] S e p KUNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY
ANTONIO RIESER AND ALEJANDRA TRUJILLO-NEGRETE
Abstract.
We prove a Kunneth theorem for the Vietoris-Rips homology andcohomology 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 thestrong graph product. We finish by computing the Vietoris-Rips cohomologyof the torus endowed with diferent semi-uniform structures. Introduction
The Vietoris-Rips complex was first defined by Vietoris in 1927 as a way to obtainhomology groups from metric spaces [11], and, somewhat later, it began to be usedin the study of hyperbolic groups [7]. With the rise of topological data analysisin 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 usedin the homological analysis of data, in addition to its natural importance in thehomological 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 notbeen established. In a previous article [9], the first author introduced a constructionof the Vietoris-Rips homology for semi-uniform spaces and proved a variant of theEilenberg-Steenrod axioms adapted to this context. In this article, we continue thedevelopment of this theory, first giving an alternate defintion of the Vietoris-Ripscomplex of a relation using simplicial sets, and then studying the Vietoris-Ripshomology and cohomology of the products of semi-uniform spaces with the goal ofestablishing K¨unneth theorems in this context. We will see that, for the Vietoris-Rips homology as defined here, although the Kunneth theorem is not true in general,we nonetheless are able to show that it does hold for semi-uniform spaces inducedby graphs. This, in turn, implies a Kunneth theorem for the classical Vietoris-Rips homology of graphs, which is the case of most interest to applications. Notethat, while it is well-known that the Kunneth theorem is false for the Vietoris-Ripshomology using Cartesian products of graphs, by translating the problem into thesetting of semi-uniform spaces, we see that that one should use the strong graphproduct instead. For a homology theory on graphs which satisfies the Kunneththeorem with the Cartesian graph product, see [6].
Centro de Investigaci´on en Matem´aticas, A. C. Jalisco S/N, Col. Valenciana CP:36023 Guanajuato, Gto, M´exico
E-mail addresses : [email protected], [email protected] .Research supported in part by Cat´edras CONACYT 1076, the US Office of Naval ResearchGlobal, and the Southern Office of Aerospace Research and Development of the US Air ForceOffice of Scientific Research. A Kunneth theorem for the classical Vietoris-Rips homology on metric spaceswith respect to the maximum metric on the product may also be deduced by apply-ing the Kunneth theorem for simplicial complexes to the isomorphism in Proposition10.2 in [1]. However, with the exception of the cases treated in [9], it remains un-clear for which cases the the classical and the semi-uniform Vietoris-Rips homologytheories coincide. There has also been some recent work on Kunneth theorems inpersistent homology [2, 5], in which an expression for the persistent homology of aproduct is obtained, given a filtered complex constructed from a category whosehomology has a Kunneth formula. The main contribution of this article is that,by constructing the Vietoris-Rips homology and cohomology in the more generalcontext of semi-uniform spaces, we are able to treat Kunneth theorems for theVietoris-Rips cohomology of graphs, metric spaces, and even topological spaces asparticular instances of the same theorem. (We refer to our earlier paper [9] forthe construction of semi-uniform structures from a topology on a space and itsassociated Vietoris-Rips homology.)2.
Semi-uniform spaces and the Vietoris-Rips complex
In this section, we recall the definition of semi-uniform spaces, which will be ourmain object of study. We begin with a few preliminary definitions.2.1.
Semi-Uniform Spaces.Definition 2.1.
Let U ⊂ X × X . We define U − := { ( y, x ) | ( x, y ) ∈ U } , and for a subset A ⊂ X , we define U [ A ] := { y ∈ X | ( a, y ) ∈ U for some a ∈ A } . 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(1) U ∈ F and U ⊂ V = ⇒ V ∈ F , and(2) U, V ∈ F = ⇒ U ∩ V ∈ F . Remark 2.3.
The condition that F does not contain the empty set is occasionallyadditional in the literature, and such filters are sometimes called proper filters . Sincewe will only be dealing with such filters, we see no need to make such a distinctionhere. Also, note that the requirement that ∅ / ∈ F combined with Condition 2 ofDefinition 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 isa 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 − contains an element of U The pair ( X, U ), consisting of a set X and a semi-uniform structure U on X , iscalled a semi-uniform space . Remark 2.5.
Note that, since U is a filter, Condition 2 in Definition 2.4 is equiv-alent to the condition that U − ∈ U . UNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY 3
We refer to [3, 8] for further details on semi-uniform spaces. We will see severalimportant examples in Sections 5 and 6, where we construct semi-uniform spacesfrom graphs and metric spaces, respectively. Additional examples can also be foundin [3, 8].2.2.
The Vietoris-Rips Homology and Cohomology of a Semi-UniformSpace.
Let ( X, U ) be a semi-uniform space and U ∈ U . We define a simplicial set X U by setting X U := { x ∈ X } = X , and, for n ∈ N , we define X Un = { ( x , . . . , x n ) | ( x i , x j ) ∈ U ) ∀ i < j } , with functions ∂ i : X Un → X Un − , where ∂ i ( x , x , ..., x n ) = ( x , x , ..., ˆ x i , ..., x n ) ,s i : X Un → X Un +1 , where s i ( x , x , ..., x n ) = ( x , x , ..., x i , x i ..., x n ) . We emphasize that the x i need not be distinct points of X , and, in particular, sincethe diagonal ∆ is a member of every U ∈ U , the X Un will contain elements with x i = x j , i = j . Definition 2.6.
We call the simplicial set X U the Vietoris-Rips complex of thepair ( X, U ).For each n ∈ Z , let C n ( X U ) be the graded free abelian group generated by theelements of the sets X Un , and let C ∗ ( X U ) = ⊕ n ∈ Z C n ( X U ). Define a differential d n : C n ( X U ) → C n − ( X U ) by d n = n X i =0 ( − i ∂ i With these definitions, ( C ∗ ( X U ) , d ) is now a chain complex, and we denote itshomology by H ∗ ( X U ).Let G be an abelian group. A q -dimensional cochain f ∈ C q ( X U ; G ) is definedas a homomorphism f : C q ( X U ) → G and the coboundary is given by( δf )( σ ) = q +1 X i =0 ( − i f ( ∂ i σ )for each ( q + 1)-simplex σ of X U . This leads to cohomology groups H ∗ ( X U , G ).We now consider maps between simplicial sets generated by different elementsof a semi-uniform structure U . Proposition 2.7.
For
U, V ∈ U , V ⊂ U , there exists a simplicial map φ V U : X V → X U . Furthermore, φ V U is an inclusion, and for W ⊂ V ⊂ U , we have φ V U ◦ φ W V = φ W U .Proof.
Since V ⊂ U , if σ = ( x , . . . , x n ) ∈ X n V , then, by definition, ( x i , x j ) ∈ V for all i < j . Therefore, ( x i , x j ) ∈ U for all i < j , and σ ∈ X Un . Define φ V U ( σ ) = σ .Since this is both a simplicial map and an inclusion, the final statement follows,and the proof is complete. (cid:3) 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 ∗ : H ∗ ( X V ) → H ∗ ( X U ) and φ ∗ UV : H ∗ ( X U ) → H ∗ ( X V ) make ( H ∗ ( X U ) , φ UV ∗ , U ) and KUNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY ( H ∗ ( X U ) , φ ∗ UV , U ) into direct and inverse systems of abelian groups, respectively.We finally define the Vietoris-Rips homology and cohomology of a semi-uniformspace ( X, U ) to be H V R ∗ ( X, U ) = lim ←− H ∗ ( X U ) H ∗ V R ( X, U ) = lim −→ H ∗ ( X U ) , respectively. We will typically suppress the U when it is unambiguous. Remark 2.8.
Note that, since the ordered and unordered simplicial complexesgive the same homology and cohomology, we note that the Vietoris-Rips homologyand cohomology defined here for a semi-uniform space ( X, U ) are isomorphic to thethose defined in [9]. 3. Products
In this section, we recall the definitions of products for semi-uniform spaces andsimplicial sets, and we then prove a theorem relating the products of Vietoris-Ripscomplexes which will be the basis for the Kunneth Theorems to follow.3.1.
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(1) { ( x, y ) | ( x, y ) ∈ X × X, a ∈ F = ⇒ ( π a x, π a y ) ∈ U a } , 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 ∆ U a ⊂ 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 wasarbitrary, ∆ X × X is contained in every element ∈ U .Suppose now that U ∈ U . Then U contains a set V of form (1) above. Since, forany U a ∈ U a we have that U − a ∈ U a , we see from V − a ⊂ U − a that V − is of theform (1) as well. However, V − ⊂ U − , so U − ∈ U , and the proof is complete. (cid:3) Definition 3.3.
We call ( X, U ) the semi-uniform product of { ( X a , U a ) } a ∈ A . Wewill sometimes denote ( X, U ) as (Π a ∈ A X a , Π a ∈ A U a ).3.2. Products of Simplicial Sets.Definition 3.4.
Let Σ and Σ ′ be simplicial sets. Then the product Σ × Σ ′ is givenby (Σ × Σ ′ ) n := { ( σ, σ ′ ) | σ ∈ Σ n , σ ′ ∈ Σ ′ n } For any simplicial sets Σ and Σ ′ , the Eilenberg-Zelber theorem [4] gives a quasi-isomorphism between the chain complexes C ∗ (Σ × Σ ′ ) and C ∗ (Σ) ⊗ C ∗ (Σ ′ ). In orderto establish the Kunneth theorems for Vietoris-Rips homology, we must furtherestablish a relationship between the Vietoris-Rips complexes of products of semi-uniform spaces on the one hand, and the products of Vietoris-Rips complexes ofsemi-uniform spaces on the other. This follows easily from the respective definitions,and is accomplished in the following theorem. First, however, we make the followingremark. UNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY 5
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 ψ : ( X × Y ) × ( X × Y ) → ( X × X ) × ( Y × Y )(( x , y ) , ( x , y )) (( x , x ) , ( y , y )Note that the expression ( X × Y ) U × V is an abuse of notation, and is, more precisely,( X × Y ) ψ − ( U × V ) . Since U × V ∼ = ψ − ( U × V ), however, we will use the first notationin 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 ( X × Y ) U × V ∼ = X U × Y V Proof.
Let φ : ( X × Y ) U × Vk → ( X U × Y V ) k be the map given by φ (( x , y ) , . . . , ( x k , y k )) := (( x , . . . , x k ) , ( y , . . . , y k )) . By definition, (( x , y ) , . . . , ( x k , y k )) ∈ ( X × Y ) U × V iff, for all i, j ∈ { , . . . , k } , oneof the following holds:(1) ψ (( x i , y j ) , ( x j , y j )) ∈ U × V (2) x i = x j and ( y i , y j ) ∈ V (3) ( x i , x j ) ∈ U and y i = y j This, in turn, is true iff ( x , . . . , x k ) ∈ X U and ( y , . . . , y k ) ∈ Y V . Therefore, φ givesan isomorphism between ( X × Y ) U × V and X U × Y V , and the proof is complete. (cid:3) The Kunneth Theorems
The Kunneth theorems for the Vietoris-Rips homology now follow from the aboveproduct relations, the Kunneth theorems for simplicial sets, and the properties ofexact sequences in direct and indirect limits. We begin with the results for Vietoris-Rips cohomology, where we have a Kunneth formula in general. In order to establishthis, we will first require the following lemma.
Lemma 4.1.
Let ( X, U ) and ( Y, V ) be semi-uniform spaces. Then for any U ∈ U , V ∈ V , and q ∈ Z , we have the short-exact sequences0 → ⊕ i + j = q H iV R ( X U ) ⊗ H jV R ( Y V ) → H qV R (( X × Y ) U × V ) →→ ⊕ i + j = q +1 Tor ( H iV R ( X U ) , H jV R ( Y V )) → Proof.
Theorem 3.5 gives C ∗ ( X × Y, U × V ) ∼ = C ∗ ( X U × Y V ), and therefore C ∗ ( X × Y, U × V ) ∼ = C ∗ ( X U × Y V ), and from the Eilenberg-Zilber theorem, we have C ∗ ( X U × Y V ) ≃ C ∗ ( X U ) ⊗ C ∗ ( Y V ), where ≃ indicates cochain-homotopy equiv-alence. The result now follows from the Kunneth theorem for cochain complexes[10], Theorem 5.4.2. (cid:3) We now give the Kunneth theorem for Vietoris-Rips cohomology.
Theorem 4.2.
Let ( X, U ) and ( Y, V ) be semi-uniform spaces, and q ∈ Z . Then → ⊕ i + j = q H iV R ( X ) ⊗ H jV R ( Y ) → H qV R ( X × Y ) →→ ⊕ i + j = q +1 Tor ( H iV R ( X ) , H jV R ( Y )) → is a short-exact sequence. KUNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY
Proof.
We first note that the family of sets in
U × V of the form U × V are cofinalin U × V . The theorem now follows from Proposition 4.1, the fact that
T or and ⊗ commute with direct limits, and that the direct limit is an exact functor. (cid:3) .For homology, the Kunneth theorem does not hold in general, due to the failureof exactness of the indirect limits of exact sequences. Nonetheless, in some specialcases, 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 4.3.
Let ( X, U ) and ( Y, V ) be semi-uniform spaces. Then for any U ∈ U , V ∈ V , and q ∈ Z , we have the short-exact sequences0 → ⊕ i + j = q H V Ri ( X U ) ⊗ H V Rj ( Y V ) → H V Rq (( X × Y ) U × V ) →→ ⊕ i + j = q − Tor ( H V Ri ( X U ) , H V Rj ( Y V )) → Proof.
Theorem 3.5 gives C ∗ ( X × Y, U × V ) ∼ = C ∗ ( X U × Y V ), and from the Eilenberg-Zilber theorem [4], we have C ∗ ( X U × Y V ) ≃ C ∗ ( X U ) ⊗ C ∗ ( Y V ), where ≃ indicateschain-homotopy equivalence. The result now follows from the Kunneth theorem forchain complexes (Theorem 5.3.3 in [10]). (cid:3) Theorem 4.4.
Let ( X, U ) and ( Y, V ) be semi-uniform spaces. Suppose U ∗ and V ∗ are maximal in U and V , respectively, ordered by inclusion. Then U ∗ × V ∗ ismaximal in U × V , and for all q ∈ Z , we have the short-exact sequences → ⊕ i + j = q H V Ri ( X ) ⊗ H V Rj ( Y ) → H V Rq ( X × Y ) →→ ⊕ i + j = q − Tor ( H V Ri ( X ) , H V Rj ( Y )) → Proof.
We first show that U ∗ × V ∗ is maximal in U × V . Suppose that there existsa W ∈ U × V with W ( U ∗ × V ∗ . Then there exist U ∈ U and V ∈ V suchthat 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, H V R ∗ ( X ) =lim ←− H V R ∗ ( X, U ). If U ∗ is maximal in U , then H ∗ ( X U ∗ ) is cofinal in the inversesystem { H ∗ ( X U ) , φ UV ∗ , U} , from which it follows that H V R ∗ ( X ) = lim ←− H V R ∗ ( X, U ) = H ∗ ( X U ∗ )as desired. Putting these together, the result now follows from Theorem 4.3. (cid:3) Although the short exact sequence doesn’t hold in general for Vietoris-Rips ho-mology, if the torsion term vanishes on a cofinal subset of the bases for the productsemi-uniform structure, we may still conclude that there is an isomorphism of therespective homology groups, as we see from the following.
Theorem 4.5.
Let ( X, U ) and ( Y, V ) be semi-uniform spaces, and suppose that F ′ ⊂ U × V is a cofinal collection of sets in of the form U × V , U ∈ U , V ∈ V such that, for any U × V ∈ F , ⊕ i + j = q − Tor ( H V Ri ( X ) , H V Rj ( Y )) = 0 . Then ⊕ i + j = q H V Ri ( X ) ⊗ H V Rj ( Y ) ∼ = H V Rq ( X × Y ) ∈ V . Kunneth Theorems for Vietoris-Rips homology on graphs
In this section, we apply the general Kunneth formulae on semi-uniform spacesfrom Section 4 to prove the Kunneth Theorem for the Vietoris-Rips homology ongraphs. We begin with the following construction. All graphs are undirected.
UNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY 7
Definition 5.1.
Let G = ( V, E ) be an graph. We denote by σ ( G ) := ( V G , E G ) thesemi-uniform space where V G = V and E G = [ E ∪ ∆ V G ], i.e. E is the filter of subsetsof V G × V G generated by the set E ∪ ∆ V G ⊂ V G × V G , where ∆ V G is the diagonalin V G × V G .Since G is undirected in the above definition, E = E − , it follows from Theorem23.A.4 in [3], that σ ( G ) is a semi-uniform space. Definition 5.2.
Let G = ( V, E ) and G ′ = ( V ′ , E ′ ) (undirected) graphs. Then thestrong graph product G ⊠ G ′ = ( V ⊠ V ′ , E ⊠ E ′ ) is defined by( v, v ′ ) ∈ V ⊠ V ′ ⇐⇒ v ∈ V, v ′ ∈ V ′ (( v , v ′ ) , ( v , v ′ )) ∈ E ⊠ E ′ ⇐⇒ One of the following holds:(1) ( v , v ) ∈ E and ( v ′ , v ′ ) ∈ E ′ (2) v = v and ( v ′ , v ′ ) ∈ E ′ (3) ( v , v ) ∈ E and v ′ = v ′ Theorem 5.3.
Let H V R ∗ ( G ) denote the Vietoris-Rips homology of the graph G =( V, E ) . Then H V R ∗ ( G ) ∼ = H V R ∗ ( σ ( G )) .Proof. By construction, E G has a maximal element E consisting of the edges of thegraph G and the diagonal ∆ V , from which it follows that H V R ∗ ( σ ( G )) ∼ = H ∗ ( V EG ).However, H ∗ ( V EG ) is exactly the homology of the simplicial set generated by theclique complex Σ G of G . Therefore, H V R ∗ ( G ) = H ∗ (Σ G ) ∼ = H ∗ ( V EG ) ∼ = H V R ∗ ( σ ( G )) , and the proof is complete. (cid:3) In order to prove the Kunneth Theorem, we will need the following
Proposition 5.4.
Let G = ( V, E ) and G ′ = ( V ′ , E ′ ) be graphs. Then ( V × V ′ , E G ⊠ G ′ ) = σ ( G ⊠ G ′ ) = σ ( G ) × σ ( G ′ ) = ( V, E G ) × ( V ′ , E G ′ ) , 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 . Bydefinition 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 E G × E G ′ ⊂ E G ⊠ G ′ .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 E G ⊠ G ′ ⊂ E G × E G ′ .It follows that E G × E G ′ = E G ⊠ G ′ , and therefore σ ( G ⊠ G ′ ) = σ ( G ) × σ ( G ). (cid:3) Theorem 5.5.
Let G = ( V, E ) and G ′ = ( V ′ , E ′ ) be graphs. For every q ∈ Z , thereexist split short exact sequences → ⊕ i + j = q H V Ri ( G ) ⊗ H V Rj ( G ′ ) → H V Rq ( G ⊠ G ′ ) →→ ⊕ i + j = q − Tor ( H V Ri ( G ) , H V Rj ( G )) → Proof.
The theorem follows immediately from Proposition 5.4, Theorem 4.4, andthe definition of the classical Vietoris-Rips complex for graphs. (cid:3)
KUNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY Applications to Metric Spaces
As an illustration of the above results, we examine the Vietoris-Rips homologyand cohomology of the torus with different semi-uniform structures. We first recallthe definition of the classical Vietoris-Rips complex on a metric space and thecorresponding relations which can be used to generate semi-uniform structures.
Definition 6.1.
Suppose that (
X, d ) is a metric space and r >
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 finitesubset σ ⊆ X is a simplex if only if the diam ( σ ) ≤ r . Definition 6.2.
Let (
X, d ) be a metric space. For every q >
0, we define U q := { ( x, y ) ⊂ X × X | d ( x, y ) < q } U ≤ q := { ( x, y ) ⊂ X × X | d ( x, y ) ≤ q } . Now fix an r >
0, and define U r to be the semi-uniform structure generated by thesets U r + ǫ for all ǫ >
0, and define U ≤ r to be the semi-uniform structure generatedby the set U ≤ r . (These are semi-uniform structures by Theorem 23.A.4 in [3]. Forother examples of common semi-uniform structures, see [3, 8].) Remark 6.3.
For a given r >
0, note that
V R < ( X ; r ) is the geometric realization ofthe simplicial set X U r , and V R ≤ ( X ; r ) is the geometric realization of the simplicialset X U ≤ r . They therefore have the same homology and cohomology groups.We now recall following theorem from [1]. Theorem 6.4 ([1], Theorem 7.4) . Denote the circle with unit circumference by S , and consider S as a metric space with the geodesic distance. For < r < ,suppose that l +1 < r < l +12 l +3 for some l ∈ { , , ... } . Then we have a homotopyequivalence (2) V R < ( S ; r ) ≃ S l +1 . The following computation now follows from Theorem 6.4 and Theorem 4.2.
Proposition 6.5.
Let < r, r ′ ≤ with l +1 < r < l +12 l +3 and l ′ +1 < r ′ < l ′ +12 l ′ +3 for some l, l ′ ∈ { , , , . . . } . Let T r,r ′ denote the product semi-uniform space T r,r ′ := ( S , U r ) × ( S , U r ′ ) .If l = l ′ , then H qV R ( T r,r ′ ) = ( Z q = 0 , l + 1 , l ′ + 1 , or l + l ′ + 1) { } otherwise.If l = l ′ , then H qV R ( T r,r ) = Z q = 0 or l + 1) Z × Z q = 2 l + 1 { } otherwise.Proof. By Theorem 4.2, we have the following short exact sequence0 → ⊕ i + j = q H iV R ( S , U r ) ⊗ H jV R ( S , U r ′ ) → H qV R ( T r,r ′ ) →→ ⊕ i + j = q − Tor ( H iV R ( S , U r ) , H jV R ( S , U r ′ )) → UNNETH THEOREMS FOR VIETORIS-RIPS HOMOLOGY 9
By Remark 6.3 and Theorem 6.4 we have that H iV R ( X, U r + ǫ ) ∼ = H i ( V R < ( X ; r + ǫ )) ∼ = H i ( S l +1 )for all ǫ > U r + ǫ are cofinal in U r , we obtain theexact sequence0 → ⊕ i + j = q H i ( S l +1 ) ⊗ H j ( S l ′ +1 ) → H V Rq ( T r,r ′ ) →→ ⊕ i + j = q − Tor ( H i ( S l +1 ) , H j ( S l ′ +1 )) → . Since the torsion term in the above exact sequence is trivial, the result follows. (cid:3)
Remark 6.6.
Note that a similar result is true for the Vietoris-Rips homology ofthe torus by Theorem 4.5.Now, suppose that (
X, d X ) and ( Y, d Y ) metric spaces and ( X × Y, d ) with d (( x , y ) , ( x , y )) = max { d X ( x , x ) , d Y ( y , y ) } . Then we have U X × Yq = { ( z, w ) ∈ ( X × Y ) × ( X × Y ) | d ( z, w ) < q } = { (( x , y ) , ( x , y )) | d (( x , y ) , ( x , y )) < q } = { (( x , y ) , ( x , y )) | max { d X ( x , x ) , d Y ( y , y ) } < q } = { (( x , y ) , ( x , y )) | d X ( x , x ) < r and d Y ( y , y ) < q }∼ = U Xq × U Yq Note, too, that the U X × Yr + ǫ = are cofinal in U X × Yr . The following corollary followsimmediately from the above comments, Theorem 4.2, and Proposition 6.5. Corollary 6.7.
Let (
X, d X ) and ( Y, d Y ) be metric spaces, and let d be the maxi-mum metric on X × Y . Then we have the short exact sequence0 → ⊕ i + j = q H iV R ( X, U r ) ⊗ H jV R ( Y, U r ) → H qV R ( X × Y, U r ) →→ ⊕ i + j = q − Tor ( H iV R ( X, U r ) , H jV R ( Y, U r )) → . Applying this to the case of the torus, we recover the following result, which alsofollows from Proposition 10.2 in [1].
Corollary 6.8.
Let T = S × S with the maximum metric. Let 0 < r ≤ with l +1 < r < l +12 l +3 for some l = 0 , , ... . Then H V Rq ( T , U r ) ∼ = ⊕ i + j = q H iV R ( S , U r ) ⊗ H jV R ( S , U r ) ∼ = Z if q ∈ { , l + 1) } Z × Z if q = 2 l + 10 otherwise. References [1] Micha l Adamaszek and Henry Adams,
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