Persistent homology of the sum metric
aa r X i v : . [ m a t h . A T ] O c t PERSISTENT HOMOLOGY OF THE SUM METRIC
GUNNAR CARLSSON, BENJAMIN FILIPPENKO
Abstract.
Given finite metric spaces (
X, d X ) and ( Y, d Y ), we investigate the per-sistent homology P H ∗ ( X × Y ) of the Cartesian product X × Y equipped with thesum metric d X + d Y . Interpreting persistent homology as a module over a polynomialring, one might expect the usual K¨unneth short exact sequence to hold. We provethat it holds for P H and P H , and we illustrate with the Hamming cube { , } k that it fails for P H n , n ≥
2. For n = 2, the prediction for P H ( X × Y ) from theexpected K¨unneth short exact sequence has a natural surjection onto P H ( X × Y ).We compute the nontrivial kernel of this surjection for the splitting of Hammingcubes { , } k = { , } k − × { , } . For all n ≥
0, the interleaving distance betweenthe prediction for
P H n ( X × Y ) and the true persistent homology is bounded aboveby the minimum of the diameters of X and Y . As preliminary results of independentinterest, we establish an algebraic K¨unneth formula for simplicial modules over thering κ [ R + ] of polynomials with coefficients in a field κ and exponents in R + = [0 , ∞ ),as well as a K¨unneth formula for the persistent homology of R + -filtered simplicialsets – both of these K¨unneth formulas hold in all homological dimensions n ≥ Contents
1. Introduction 22. κ [ R + ]-modules 62.1. Finiteness and Freeness 72.2. Bars 92.3. Simplicial κ [ R + ]-modules 102.4. The Algebraic Persistent K¨unneth Theorem 103. R + -filtered simplicial sets 114. Metric spaces 134.1. Persistent homology 144.2. A long exact sequence for the sum metric d X + d Y d X + d Y in low dimensions 174.4. Interleaving distance 204.5. Example: The Hamming Cube I k Introduction
Topological Data Analysis (TDA) aims to understand the topology of an ambientspace from a finite sample; see [2] for an overview of the field. We assume that thetopology of the ambient space is induced by a metric d , making the finite sample X into a finite metric space ( X, d ). In applications, the ambient space is often embeddedin R n and various metrics d on R n can be useful.A central tool in TDA is the persistent homology P H ∗ ( X ) of the Vietoris-Ripscomplex associated to a finite metric space ( X, d ). Persistent homology providesapproximations of the homology of the ambient space from which X was sampled, andit does so at all scales t ≥
0. The scale parameter t takes all values t ∈ R + = [0 , ∞ ),and for each fixed value it upper bounds the distances in X that are ‘seen at scale t .’Many applications now use persistent homology for feature generation. This ideais used, for example, in the study of databases of molecules [9]. The results of thepresent paper give relationships among such features, which can be expected to beuseful as the applications of the method become more sophisticated.In this paper, given two finite metric spaces ( X, d X ) and ( Y, d Y ), we equip theCartesian product X × Y with the sum metric d X + d Y . We are interested in computing P H ∗ ( X × Y ) in terms of P H ∗ ( X ) and P H ∗ ( Y ). A K¨unneth formula of this typewould allow us to compute persistent homology of interesting spaces, such as theHamming cube I k for k ≥ , I = { , } with the Hamming distance; see Example 1.7and Section 4.5 for a discussion of the Hamming cube. In [5], the authors derive aK¨unneth formula for the maximum metric d X × Y := max { d X , d Y } on X × Y whichholds in all homological dimensions.Our main result (Theorem 1.1) is that the familiar K¨unneth short exact sequenceholds for the sum metric d X × Y := d X + d Y in homological dimensions 0 and 1, andin dimension 2 it computes a module that admits a surjection onto P H ( X × Y );see Theorem 4.5 for a more refined statement and a proof. Example 1.7 below showsthat the short exact sequence fails in homological dimension n ≥
2. We discussthe underlying reason for this failure below (both in this introduction and in thebeginning of Section 4). Here we view persistent homology
P H ∗ ( X ) as a module overa ring κ [ R + ] of polynomials in a single variable with exponents in R + = [0 , ∞ ) andcoefficients in a field κ ; see Section 2 for a study of κ [ R + ]-modules, and in particularsee Remark 2.1 for a discussion of their relationship with persistence vector spaces. Theorem 1.1.
Consider finite metric spaces ( X, d X ) and ( Y, d Y ) and the productspace equipped with the sum metric ( X × Y, d X + d Y ) . Then for n = 0 , there is ashort exact sequence → M i + j = n P H i ( X ) ⊗ κ [ R + ] P H j ( Y ) → P H n ( X × Y ) → M i + j = n − T or ( P H i ( X ) , P H j ( Y )) → which is natural with respect to distance non-increasing maps ( X, d X ) → ( X ′ , d X ′ ) and ( Y, d Y ) → ( Y ′ , d Y ′ ) . This sequence splits, but not naturally. Moreover, in dimension n = 2 , the direct sum of the term in the left position (tensor product terms) and theterm in the right position (Tor terms) admits a surjection onto P H ( X × Y ) . (cid:3) ERSISTENT HOMOLOGY OF THE SUM METRIC 3
Remark 1.2.
To compute the bars in
P H n ( X × Y ) for n = 0 , using the split shortexact sequence in Theorem 1.1, it suffices to use the formulas for tensor product andTor of bars computed in Proposition 2.6. See Section 2.2 for an explanation of bars. Remark 1.3.
Theorem 1.1 and all other results about metric spaces ( X, d ) in thispaper hold for a more general class of spaces: Precisely, it is required that the pairing d : X × X → R + is symmetric d ( x, y ) = d ( y, x ) , but d is not required to be reflexive d ( x, y ) = 0 ⇐⇒ x = y or satisfy the triangle inequality d ( x, z ) ≤ d ( x, y ) + d ( y, z ) . For all n ≥
0, denote the prediction for
P H n ( X × Y ) from the short exact sequencein Theorem 1.1 by P H n ( X, Y ). That is,
P H n ( X, Y ) is isomorphic to the direct sumof the tensor product terms on the left and the Tor terms on the right. The followingresult bounds the interleaving distance between the homology
P H n ( X × Y ) and theprediction P H n ( X, Y ) from above by the minimum of the diameters of X and Y ; seeTheorem 4.9 for the proof. We discuss the interleaving distance in Section 4.4. Theidea to look for a bound on this distance was suggested by Leonid Polterovich. Theorem 1.4.
The interleaving distance between
P H ∗ ( X, Y ) and P H ∗ ( X × Y ) isless than or equal to the minimum of the diameters min(diameter( X ) , diameter( Y )) , where the diameter is the maximum distancediameter ( X ) = max { d X ( x , x ) | x , x ∈ X } . (cid:3) We now outline the structure of the paper, including preliminary K¨unneth typeresults used in the proof of Theorem 1.1 which are of independent interest. Theseother K¨unneth formulas hold in all homological dimensions n ≥
0. We also summarizewhy Theorem 1.1 holds, and why it fails in homological dimensions n ≥ κ [ R + ]-modules and simplicial modules(Defintion 2.7). Given a simplicial κ [ R + ]-module M , there is an associated chaincomplex C ∗ ( M ); see Definition 2.8. The main result (Theorem 1.5) proved in Sec-tion 2.4 is an algebraic K¨unneth formula for the homology H ∗ ( C ∗ ( M ⊗ κ [ R + ] N )) forfree finitely generated simplicial κ [ R + ]-modules M and N . See Theorem 2.10 for theproof. We remark that, if κ [ R + ] were a principal ideal domain (PID), then this theo-rem would be immediate from the usual K¨unneth short exact sequence for simplicialmodules over a PID. However, unlike the usual polynomial ring κ [ Z + ] with exponentsin Z + = { , , , . . . } , the ring κ [ R + ] is not a PID. Related K¨unneth formulas havebeen proven in [7] and [1]. Theorem 1.5. (Algebraic Persistent K¨unneth Theorem)
Let
M, N be R + -graded free finitely generated simplicial κ [ R + ] -modules. Then for n ≥ there is ashort exact sequence → M i + j = n H i ( C ∗ ( M )) ⊗ κ [ R + ] H j ( C ∗ ( N )) → H n ( C ∗ ( M ⊗ κ [ R + ] N )) → M i + j = n − T or ( H i ( C ∗ ( M )) , H j ( C ∗ ( N ))) → GUNNAR CARLSSON, BENJAMIN FILIPPENKO which is natural in both M and N . Moreover, the sequence splits, but not naturally. (cid:3) Building on this algebraic K¨unneth formula over κ [ R + ], in Section 3 we establish aK¨unneth formula (Theorem 1.6) for persistent homology of R + -filtered simplicial sets( X , l ) (Definitions 3.1, 3.3). These are simplicial sets X (see Section 5 for a review ofsimplicial sets) together with a map l : X n → R + on the n -simplices X n of X for each n ≥ l is non-increasing along the structure maps (face and degeneracy)in the simplicial set. For any t ∈ R + , the collection of simplices σ ∈ X such that l ( σ ) ≤ t forms a simplicial set X ( t ) , and if t ≤ t ′ then X ( t ) ⊆ X ( t ′ ) . The persistenthomology P H ∗ ( X , l ) is a R + -graded κ [ R + ]-module with homogeneous degree- t partequal to the homology of the simplicial set X ( t ) , i.e. P H ( t ) ∗ ( X , l ) = H ∗ ( X ( t ) ; κ ) . The product of filtered simplicial sets ( X , l X ) and ( Y , l Y ) is the pair ( X × Y , l X + l Y )where X × Y is the usual product of simplicial sets and l X + l Y is the pointwise sumof the filtration functions. In the notation of the above paragraph, this correspondsto the filtration ( X × Y ) ( t ) = S t X + t Y ≤ t X ( t X ) × Y ( t Y ) for t ∈ R + . The followingK¨unneth theorem follows from Theorem 1.5; see Section 3 for the proof. In [5], theauthors derive K¨unneth formulas for the persistent homology of the product X × Y of Z + -filtered topological spaces X = ( X (0) ⊂ X (1) ⊂ X (2) ⊂ · · · ) for various choicesof filtration on X × Y , including (
X × Y ) ( t ) = S t X + t Y ≤ t X ( t X ) × Y ( t Y ) for t ∈ Z + . Theorem 1.6. (K¨unneth Theorem for R + -filtered simplicial sets) Given R + -filtered simplicial sets ( X , l X ) and ( Y , l Y ) such that X n and Y n are finite sets for all n ≥ , then for n ≥ there is a short exact sequence → M i + j = n P H i ( X , l X ) ⊗ κ [ R + ] P H j ( Y , l Y ) → P H n ( X × Y , l X + l Y ) → M i + j = n − T or ( P H i ( X , l X ) , P H j ( Y , l Y )) → which is natural with respect to maps ( X , l X ) → ( X ′ , l X ′ ) and ( Y , l Y ) → ( Y ′ , l Y ′ ) .Moreover, the sequence splits, but not naturally. (cid:3) In Section 4 we study the persistent homolgy
P H ∗ ( X ) of metric spaces ( X, d X ).We define P H ∗ ( X ) to be the persistent homology P H ∗ ( X , l X ) of a R + -filtered sim-plicial set ( X , l X ) associated to the metric space. The filtration function l X is definedusing the metric d X (see (10)). The homogeneous degree- t part P H ( t ) ∗ ( X ) is equal tohomology H ∗ ( X ( t ) ; κ ), which in turn is isomorphic to homology of the time- t Vietoris-Rips simplicial complex V ( X, t ) since V ( X, t ) and X ( t ) are homotopy equivalent; seeRemark 4.2 for Vietoris-Rips V ( X, t ). That is, we have
P H ( t ) ∗ ( X ) ∼ = H ∗ ( V ( X, t ); κ )for all t ∈ R + .The product of two metric spaces X × Y equipped with the sum metric d X + d Y has associated R + -filtered simplicial set ( X × Y , l X × Y ), where the filtration function l X × Y is defined using the sum metric d X + d Y . The underlying reason for the failure ofTheorem 1.1 in dimensions n ≥ l X × Y ≤ l X + l Y , ERSISTENT HOMOLOGY OF THE SUM METRIC 5 but equality does not necessarily hold. The inequality provides a graded modulehomomorphism
P H n ( X × Y , l X + l Y ) → P H n ( X × Y , l X × Y ) = P H n ( X × Y ) , but since equality does not hold, it is not necessarily an isomorphism. If it were anisomorphism, then Theorem 1.1 would follow immediately from Theorem 1.6 in alldimensions n ≥
0. We show in Section 4.2 that this homomorphism fits into a longexact sequence, and in Section 4.3 we show that the correction term in the long exactsequence vanishes in homological dimensions n = 0 , ,
2. Hence the homomorphismis an isomorphism for n = 0 , n = 2. Theorem 1.1 then followsfrom Theorem 1.6.We remark that, in the notation of Theorem 1.4, we have P H n ( X, Y ) =
P H n ( X × Y , l X + l Y )by definition; see (11) and Proposition 4.3.In Section 4.5 and Example 1.7, we investigate P H ∗ ( I k ) where I k = { , } k is theHamming cube with the Hamming metric. We use the splitting I k = I k − × I ofthe Hamming metric to apply Theorem 1.1 inductively to compute P H n ( I k ) for n =0 , , , and for all k ≥
1. This involves a technical computation of the nontrivial kernelof the surjection
P H ( I × I k − , l I + l I k − ) → P H ( I k ); see Propositions 4.12, 4.13. Example 1.7.
Consider the metric space I = { , } with distance between itstwo points. The k -dimensional Hamming Cube is the k -fold Cartesian product I k consisting of k -tuples of zeros and ones equipped with the Hamming metric: Thedistance between two k -tuples in I k is the number of coordinates in which they are notequal. The Hamming metric is equal to the sum metric on I k given by summing themetric on each of the k factors of I , which is also equal to the sum metric on the twofactor splitting I k = I k − × I . So, from a full K¨unneth formula for the sum metricon a product X × Y , one hopes to inductively compute P H n ( I k ) for all n and k .In Section 4.5 we prove that P H ( I k ) = 0 for all k . This shows that Theorem 1.1does not hold in dimension n = 2 . Indeed, for X = I and Y = I , the product is X × Y = I , and the short exact sequence in Theorem 1.1 for n = 2 would predict that P H ( I ) has bar; see Section 2.2 for a review of bars and how to compute tensorproduct and Tor of bars.The table below displays the number of bars in P H n ( I k ) for low values of n and k .This data shows that Theorem 1.1 does not hold in higher dimensions n > . Indeed,again for X = I and Y = I with product X × Y = I , the short exact sequence inTheorem 1.1 for n = 3 would predict that P H ( I ) = 0 , but in fact P H ( I ) has bar. GUNNAR CARLSSON, BENJAMIN FILIPPENKO
Number of bars in
P H n ( I k ) k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 P H ( I k ) P H ( I k ) P H ( I k ) P H ( I k ) P H ( I k ) P H ( I k ) P H ( I k ) P H ( I k ) (cid:3) κ [ R + ] -modules The ring κ [ R + ] is the monoid ring of R + = [0 , ∞ ) over a field κ . Concretely, it isthe polynomial ring in a single variable T with exponents in R + and coefficients in κ . There is a natural R + -grading on κ [ R + ] where the homogeneous degree t ∈ R + elements are the monomials with exponent t . The usual polynomial ring in a singlevariable is the monoid ring κ [ Z + ] over the nonnegative integers Z + = { , , , . . . } ,i.e., the exponents on the variable are in Z + .We are interested in the ring κ [ R + ] because the persistent homology P H ∗ ( X ) of ametric space ( X, d ) is a κ [ R + ]-module. Indeed, P H ∗ ( X ) is often viewed as a persis-tence vector space – or in other words a functor from the poset R + to the category of κ -vector spaces – and a persistence vector space is equivalent to a R + -graded κ [ R + ]-module; see Remark 2.1. For the purposes of this paper, it is more convenient towork with κ [ R + ]-modules.Our main result in this section in an algebraic K¨unneth formula (Theorem 2.10)for simplicial modules over κ [ R + ] (Section 2.3). Theorem 2.10 would be immediatefrom the usual K¨unneth short exact sequence for simplicial modules over a principalideal domain – but κ [ R + ] is not a principal ideal domain, unlike the usual polynomialring κ [ Z + ] with integer exponents. Indeed, κ [ R + ] is not even Noetherian: Considerthe ascending chain of principle ideals ( T ) ⊂ ( T / ) ⊂ ( T / ) ⊂ · · · . We resolve theseissues in Section 2.1: The ring κ [ R + ] has the crucial property that the kernel andthe image of a homomorphism between free finitely generated modules are both freeand finitely generated; see Lemma 2.4. This property of κ [ R + ] is enough to establishTheorem 2.10.In Section 2.2, we discuss the barcode classification of finitely presented persistencevector spaces in the language of finitely presented κ [ R + ]-modules (Proposition 2.5),and we compute the tensor product and Tor of bars in Proposition 2.6. This allowsfor efficient computation with the algebraic K¨unneth theorem (Theorem 2.10). Remark 2.1. (Persistence vector spaces and Artin-Rees)
The Artin-Rees con-struction provides an equivalence between finitely generated R + -graded κ [ R + ] -modulesand finitely generated persistence vector spaces over κ . We describe this equivalencein this remark. See [10, Theorem 3.1] for a discussion over Z + instead of R + . ERSISTENT HOMOLOGY OF THE SUM METRIC 7
Recall the definition of a persistence vector space V = { V t } t ∈ R + over κ from [2,Def. 3.3] . This is a functor V : R + → V ec κ , where here R + denotes the partiallyordered set R + viewed as a category and V ec κ is the category of vector spaces over κ .Let P V ec κ denote the category of persistence vector spaces over κ .The Artin-Rees construction provides an equivalence of categories A : P V ec κ → κ [ R + ] -Mod, where κ [ R + ] -Mod is the category of graded κ [ R + ] -modules. Explicitly, A ( V ) is the κ -vector space ⊕ t ∈ R + V t . The κ [ R + ] -module structure on A ( V ) is given asfollows. Let T denote the variable in κ [ R + ] , and for t ≤ t ′ let l t,t ′ : V t → V t ′ denotethe corresponding map in V . Then for a homogeneous element v ∈ V t we define T a · v = l t,t + a ( v ) ∈ V t + a for a ∈ R + . This gives A ( V ) the structure of a R + -graded κ [ R + ] -module.The inverse functor A − : κ [ R + ] -Mod → P V ec κ is as follows. For M ∈ κ [ R + ] -Modand t ∈ R + , the κ -vector space A − ( M ) t is the homogeneous degree t part of M . For t ≤ t ′ the linear map l t,t ′ : A − ( M ) t → A − ( M ) t ′ is multiplication by T t ′ − t . Finiteness and Freeness.
Over the ring κ [ R + ], submodules of finitely gener-ated free modules are not necessarily finitely generated free: Indeed, the ideal in κ [ R + ]consisting of all polynomials with zero constant term is neither free nor finitely gen-erated. This is in stark contrast to the polynomial ring with integer exponents κ [ Z + ]which is a principal ideal domain and hence does have the property that submodulesof finitely generated free modules are always finitely generated free.To prove the algebraic K¨unneth formula over κ [ R + ] (Theorem 2.10) and to de-scribe homology using bars via the classification theorem for finitely presented κ [ R + ]-modules (Proposition 2.5), it is essential that, given a homomorphism of finitelygenerated graded free κ [ R + ]-modules, the kernel and the image are finitely generatedand graded free. This is the content of Lemma 2.4. Remark 2.2.
The following results, mostly in this subsection, are a rephrasing ofresults in [2, Sec. 3.4] from the language of persistence vector spaces to the languageof κ [ R + ] -modules (see Remark 2.1 for a discussion of the equivalence between thesenotions): Lemma 2.3 corresponds to [2, Prop. 3.7] . Lemma 2.4 corresponds to theessential step in the proof of [2, Prop. 3.12] . And in Proposition 2.5, we reinterpretthe classification of finitely presented persistence vector spaces [2, Prop. 3.12, 3.13] asa classification of finitely presented R + -graded κ [ R + ] -modules. To prepare for Lemma 2.4, we first understand the matrix of a R + -graded κ [ R + ]-module homomorphism with respect to a basis. Lemma 2.3.
Consider free and finitely generated R + -graded κ [ R + ] -modules P and Q with bases { p , . . . , p n } and { q , . . . , q m } , respectively, such that each p j and q i ishomogeneous.Then, given a R + -graded κ [ R + ] -module homomorphism ϕ : P → Q , its matrix A ϕ with respect to the given bases has homogeneous entries satisfying (1) deg( A ϕi,j ) = deg( p j ) − deg( q i ) ≥ or A ϕi,j = 0 for all i, j. Conversely, any ( m × n ) -matrix A with homogeneous entries in κ [ R + ] satisfying (1) determines a R + -graded κ [ R + ] -module homomorphism ϕ A : P → Q by ϕ A ( p j ) = P mi =1 A i,j q i .Moreover, the correspondences ϕ A ϕ and A ϕ A are inverses of each other. GUNNAR CARLSSON, BENJAMIN FILIPPENKO
Proof.
Consider a R + -graded κ [ R + ]-module homomorphism ϕ : P → Q . Then, theentries of the matrix A ϕ are defined by ϕ ( p j ) = P mi =1 A ϕi,j q i . Since ϕ is R + -graded,the image ϕ ( p j ) is homogeneous of degree deg( p j ). Since the q i form a basis of Q , thisimplies that either A ϕi,j q i is 0 or it is homogeneous of degree deg( p j ) for all i , whichimplies that either A ϕi,j = 0 or deg( A ϕi,j ) = deg( p j ) − deg( q i ) for all i , as claimed. Theconverse statement is clear. (cid:3) Lemma 2.4.
Suppose ϕ : P → Q is a homomorphism of R + -graded free finitelygenerated κ [ R + ] -modules.Then, there exist bases { p , . . . , p n } of P and { q , . . . , q m } of Q consisting of homo-geneous elements such that the matrix of ϕ with respect to these bases has the propertythat every row and every column has at most one nonzero entry.In particular, the kernel and the image of ϕ are graded free and finitely generated.Proof. Since P and Q are graded free and finitely generated, there exists a basis { p , . . . , p n } of P and a basis { q , . . . , q m } of Q , both consisting of homogeneouselements. Let A be the matrix of ϕ with respect to these bases. We must show thatthere exist invertible matrices B and C satisfying (1) such that the matrix BAC hasat most one nonzero entry in each row and column; precisely, (1) requires that B satisfies deg( B i,j ) = deg( q j ) − deg( q i ) ≥ B i,j = 0 for all 1 ≤ i, j ≤ m and that C satisfies the same condition with respect to the p i .Let q i be the basis element such that deg( q i ) is maximal among the degrees of the q i such that the i -th row of A is nonzero, and let p j be the basis element such thatdeg( p j ) is minimal among the degrees of the p j such that A i ,j = 0. By Lemma 2.3,we have A i ,j = l · T deg( p j ) − deg( q i ) for some l ∈ κ . For any j = j such that A i ,j = 0, we have A i ,j = l · T deg( p j ) − deg( q i ) for some l ∈ κ . There is a column operation given by a matrix C with C j ,j = − ( l/l ) · T deg( p j ) − deg( p j ) , ones on the diagonal, and zeros elsewhere, and we have( AC ) i ,j = 0. Applying these column operations for all j = j such that A i ,j = 0arrives at a matrix with i -th row equal to 0 except for the ( i , j )-entry. Rename C to be the product of all these column operations, and set A ′ = AC . Now, for any i = i such that A ′ i,j = 0, we have A ′ i,j = l · T deg( p j ) − deg( q i ) for some l ∈ κ . Thereis a row operation given by a matrix B with B i,i = − ( l/l ) · T deg( q i ) − deg( q i ) , oneson the diagonal, and zeros elsewhere, and we have ( BA ′ ) i,j = 0 . Applying these rowoperations for all i = i such that A ′ i,j = 0 arrives at a matrix with j -th columnequal to 0 except for the ( i , j )-entry. Rename B to be the product of all these rowoperations. Then the matrix BAC has the property that the i -th row and the j -thcolumn are zero except for the ( i , j )-entry.We may now iterate this process to produce the desired result. Indeed, relabel BAC by A , let q i denote the basis element such that deg( q i ) is maximal among thedegrees of the q i = q i such that the i -th row of A is nonzero, and let p j denote thebasis element such that deg( p j ) is minimal among the degrees of the p j = p j suchthat A i ,j = 0. Then the row and column operations as described above that zero outall entries in the i -row and j -column (except for the ( i , j )-entry) do not affect the i -row and the j -column. Repeating this process completes the construction. (cid:3) Graded free means that there exists a basis of homogeneous elements.
ERSISTENT HOMOLOGY OF THE SUM METRIC 9
Bars.
A common notation for finitely presented κ [ R + ]-modules uses bars ( a, b )for 0 ≤ a ≤ b ≤ ∞ , where ( a, b ) is shorthand for the module generated by a singlegenerator σ of degree a such that T b − a · σ = 0; precisely,( a, b ) = Γ a κ [ R + ] /T b − a , where Γ a denotes a degree shift by a . We allow b = ∞ , in which case( a, ∞ ) = Γ a κ [ R + ] . The following classification result is equivalent to the classification of finitely pre-sented persistence vector spaces in [2, Prop. 3.12, 3.13]; see Remark 2.2. The essentialstep in the setting of κ [ R + ]-modules is Lemma 2.4. Proposition 2.5.
Every R + -graded finitely presented κ [ R + ] -module M is isomorphicto a direct sum of the form M ∼ = ⊕ ni =1 ( a i , b i ) , where ≤ a i < b i ≤ ∞ . Moreover, this decomposition is unique, i.e. if M isisomorphic to another such direct sum then the number n of bars is the same and theset of pairs ( a i , b i ) that occur, with multiplicities, is the same. (cid:3) Due to the above classification, for many computations it suffices to understand howsingle bars interact with each other. In this paper we use the following computationsof the tensor product and
T or of bars; equivalent computations appear in [1][7].
Proposition 2.6.
Consider the bars ( a, b ) and ( c, d ) . Then there are isomorphismsof R + -graded κ [ R + ] -modules (2) ( a, b ) ⊗ κ [ R + ] ( c, d ) ∼ = ( a + c, min { a + d, b + c } ) and (3) T or (( a, b ) , ( c, d )) ∼ = (max { a + d, b + c } , b + d ) . Moreover, for any R + -graded finitely presented κ [ R + ] -modules M and N , we have T or n ( M, N ) = 0 for n ≥ .Proof. We compute ( a, b ) ⊗ κ [ R + ] ( c, d ) = Γ a κ [ R + ] /T b − a ⊗ Γ c κ [ R + ] /T d − c ∼ = Γ a + c κ [ R + ] / h T b − a , T d − c i = Γ a + c κ [ R + ] /T min { b − a,d − c } = ( a + c, a + c +min { b − a, d − c } ) =( a + c, min { a + d, b + c } ) , as claimed.Next, we compute (3). A projective resolution of ( a, b ) is given by the exact se-quence(4) 0 → ( b, ∞ ) → ( a, ∞ ) → ( a, b ) → . Removing the ( a, b ) term and tensoring with ( c, d ) yields the complex0 → ( b + c, b + d ) → ( a + c, a + d ) → . The homology at the ( b + c, b + d ) term is the kernel of the map to ( a + c, a + d ).If b + c ≥ a + d then the map is 0 and hence the homology is ( b + c, b + d ). If b + c ≤ a + d then the kernel is ( a + d, b + d ). So, in both cases, the homology is(max { a + d, b + c } , b + d ), proving (3).To prove the final statement of the proposition, first apply Proposition 2.5 to both M and N to write them as direct sums of bars. Then the statement follows since theprojective resolution (4) is zero to the left of the ( b, ∞ ) term. (cid:3) Simplicial κ [ R + ] -modules. Let κ [ R + ]-Mod denote the category with objectsthe R + -graded κ [ R + ]-modules and maps the graded module homomorphisms. Definition 2.7. A R + -graded simplicial κ [ R + ] -module M is a simplicial objectin κ [ R + ] -Mod, i.e. a contravariant functor M : ∆ → κ [ R + ] -Mod, where ∆ is thesimplex category (see Section 5). Concretely, M is a collection of R + -graded κ [ R + ]-modules M n = M ([ n ]) for n ≥ R + -graded κ [ R + ]-module maps M ϕ : M n → M m for every map ϕ :[ m ] → [ n ] in ∆ that together satisfy the usual functorial properties. We call M n the n -simplices of M . The images M d ∆i and M s ∆i of the coface maps d ∆i and thecodegeneracy maps s ∆i are called face maps and degeneracy maps, respectively, andfor convenience we often denote them by d i and s i .There is a natural chain complex associated to a simplicial module. Definition 2.8.
The alternating face maps complex C ∗ ( M ) associated to agraded simplicial κ [ R + ] -module M is a chain complex of R + -graded κ [ R + ] -moduleswhich in level n ≥ is the module C n ( M ) = M n and where the differential is given by the alternating sum of the face maps, i.e. d : C n +1 ( M ) → C n ( M ) x n +1 X i =0 ( − i d i ( x ) . The Algebraic Persistent K¨unneth Theorem.
For graded simplicial κ [ R + ]-modules M and N (Definition 2.7), the tensor product M ⊗ N is defined level-wise,i.e. ( M ⊗ N ) n := M n ⊗ κ [ R + ] N n for n ≥
0. For a morphism ϕ : [ m ] → [ n ] in ∆ , the corresponding map in M ⊗ N isgiven by ( M ⊗ N ) ϕ : M n ⊗ N n → M m ⊗ N m x ⊗ y M ϕ ( x ) ⊗ N ϕ ( y ) . We establish in Theorem 2.10 a K¨unneth formula for the homology of C ∗ ( M ⊗ N )in terms of C ∗ ( M ) and C ∗ ( N ), where C ∗ ( − ) is the alternating face maps complex(Definition 2.8).In level n ≥
0, we have C n ( M ⊗ N ) = C n ( M ) ⊗ κ [ R + ] C n ( N ) . There is another chain complex of κ [ R + ]-modules associated to the pair M, N, givenby the usual tensor product of chain complexes C ∗ ( M ) ⊗ C ∗ ( N ), which in level n isthe κ [ R + ]-module ( C ∗ ( M ) ⊗ C ∗ ( N )) n = M i + j = n C i ( M ) ⊗ κ [ R + ] C j ( N ) . ERSISTENT HOMOLOGY OF THE SUM METRIC 11
The Eilenberg-Zilber theorem (see [4], [8, Thm. 8.5.1]) says that there is a naturalisomorphism on homology(5) H ∗ ( C ∗ ( M ) ⊗ C ∗ ( N )) ∼ = H ∗ ( C ∗ ( M ⊗ N )) . Definition 2.9.
A simplicial module M is graded free (resp. finitely generated )if, for all n ≥ , the module M n is graded free (resp. finitely generated). We now prove Theorem 1.5 from the introduction.
Theorem 2.10. (Algebraic Persistent K¨unneth Theorem)
Let
M, N be R + -graded free finitely generated simplicial κ [ R + ] -modules. Then for n ≥ there is ashort exact sequence → M i + j = n H i ( C ∗ ( M )) ⊗ κ [ R + ] H j ( C ∗ ( N )) → H n ( C ∗ ( M ⊗ κ [ R + ] N )) → M i + j = n − T or ( H i ( C ∗ ( M )) , H j ( C ∗ ( N ))) → which is natural in both M and N . Moreover, the sequence splits, but not naturally.Proof. For n ≥
0, the submodules B n ( M ) and Z n ( M ) of C n ( M ) = M n consisting ofboundaries and cycles, respectively, are graded free by Lemma 2.4, and similarly forthe boundaries and cycles in C n ( N ). Hence, although κ [ R + ] is not a principal idealdomain, the proof of the K¨unneth short exact sequence for the homology of the tensorproduct of chain complexes C ∗ ( M ) ⊗ C ∗ ( N ) still goes through; see for example theproof of [6, Theorem 3B.5]. That is, for n ≥ → M i + j = n H i ( C ∗ ( M )) ⊗ κ [ R + ] H j ( C ∗ ( N )) → H n ( C ∗ ( M ) ⊗ C ∗ ( N )) → M i + j = n − T or ( H i ( C ∗ ( M )) , H j ( C ∗ ( N ))) → (cid:3) We can compute H n ( C ∗ ( M ⊗ κ [ R + ] N )) using the K¨unneth short exact sequence inTheorem 2.10 together with the classification result Proposition 2.5 and the calcula-tions of tensor product and Tor of bars in Proposition 2.6.3. R + -filtered simplicial sets We define a notion of a simplicial set filtered by R + = [0 , ∞ ) (see Section 5 for areview of simplicial sets). Its homology is a R + -graded module over the ring κ [ R + ],whose theory we discuss in Section 2. In Section 4 we associate a R + -filtered simplicialset to a metric space ( X, d ) and we define persistent homology of the metric space
P H ∗ ( X ) to be the homology of the filtered simplicial set. This is a rephrasing of theVietoris-Rips construction; see Remark 4.2.In this section we prove the K¨unneth theorem for R + -filtered simplicial sets (The-orem 1.6), which follows easily from the algebraic K¨unneth theorem for simplicial κ [ R + ]-modules (Theorem 2.10). Definition 3.1. A R + -filtered simplicial set ( X , l ) is a simplicial set X togetherwith maps l : X n → R + for all n ≥ satisfying l ( X ϕ ( σ )) ≤ l ( σ ) for all simplices σ ∈ X n and all morphisms ϕ : [ m ] → [ n ] in ∆ . Remark 3.2.
A natural notion of a R + -filtered simplicial set would be a collectionof simplicial sets { X ( t ) } t ∈ R + and inclusion maps X ( t ) ֒ → X ( t ′ ) for t ≤ t ′ . Given a R + -filtered simplicial set ( X , l ) as in Definiton 3.1, for any t ∈ R + , the collection ofsimplices σ ∈ X ∗ such that l ( σ ) ≤ t forms a simplicial set X ( t ) , and if t ≤ t ′ thenthere is an inclusion X ( t ) ֒ → X ( t ′ ) .The persistent homology P H ∗ ( X , l ) (see Definition 3.3) is a R + -graded κ [ R + ] -module with homogeneous degree- t part equal to the homology of the simplicial set X ( t ) , i.e. P H ( t ) ∗ ( X , l ) = H ∗ ( X ( t ) ; κ ) . Let R + -sSet denote the category with objects the R + -filtered simplicial sets ( X , l )and the morphisms f : ( X , l ) → ( X ′ , l ′ ) given by the simplicial maps X → X ′ satisfying l ′ ( f ( σ )) ≤ l ( σ ) for all σ ∈ X . Let κ [ R + ]-sMod denote the category of R + -graded simplicial κ [ R + ]-modules (Def-inition 2.7).We now define a functor(6) F : R + -sSet → κ [ R + ]-sModand then define the persistent homology of ( X , l ) (Definition 3.3) to be the homologyof F ( X , l ). Let the n -simplices of F ( X , l ) be the free module(7) F n ( X , l ) := M σ ∈ X n Γ l ( σ ) κ [ R + ] · h σ i , where the notation Γ a means a degree shift of a ∈ R + . So, F is essentially the free κ [ R + ]-module functor, with additional degree shifts of the generators by the filtrationfunction l . For a morphism ϕ : [ m ] → [ n ] in ∆ , we define the corresponding morphismin F ( X , l ) on generators by F ( X , l ) ϕ : F n ( X , l ) → F m ( X , l ) h σ i 7→ T l ( σ ) − l ( X ϕ ( σ )) h X ϕ ( σ ) i . Extending κ [ R + ]-linearly defines F ( X , l ) ϕ on all of F n ( X , l ). Note that F ( X , l ) ϕ is agraded map due to the degree shifts in (7). In particular, the face maps on F ( X , l )are given on generators by d i ( h σ i ) := T l ( σ ) − l ( d i ( σ )) h d i ( σ ) i , where on the left we are writing d i = F ( X , l ) d ∆i and on the right we are writing d i = X d ∆i . Alternatively, we may define R + -sSet as follows. Consider the category of pairs ( A, l ) of sets A and maps l : A → R + , where the morphisms f : ( A, l ) → ( A ′ , l ′ ) are the set maps f : A → A ′ suchthat l ′ ( f ( a )) ≤ l ( a ) for all a ∈ A . Such a pair ( A, l ) is called a R + -filtered set, as defined in [2,Sec. 3.4], and we denote the category of these pairs by R + -Set. Then a R + -filtered simplicial set isa simplicial object in R + -Set, i.e. a contravariant functor ∆ → R + -Set, and R + -sSet is the categoryof these functors. ERSISTENT HOMOLOGY OF THE SUM METRIC 13
For a map f : ( X , l ) → ( X ′ , l ′ ) in R + -sSet, we define F ( f ) on generators by F ( f ) : F n ( X , l ) → F n ( X ′ , l ′ ) h σ i 7→ T l ( σ ) − l ′ ( f ( σ )) h f ( σ ) i . Note that the property l ( σ ) − l ′ ( f ( σ )) ≥
0, which holds by definition of a map f ∈ R + -sSet, is crucial in the definition of F ( f ); indeed, the factor T l ( σ ) − l ′ ( f ( σ )) isnecessary to make F ( f ) a R + -graded map due to the degree shifts in (7). Definition 3.3.
Given ( X , l ) ∈ R + -sSet, its persistent chain complex P C ∗ ( X , l ) is the alternating face maps complex (Defintion 2.8) of the module F ( X , l ) , P C ∗ ( X , l ) := C ∗ ( F ( X , l )) , and its persistent homology is the homology P H ∗ ( X , l ) := H ∗ ( P C ∗ ( X , l )) . For t ∈ R + , the time- t persistent homology P H ( t ) ∗ ( X , l ) is the homogeneous degree t part of P H ∗ ( X , l ) . Consider R + -filtered simplicial sets ( X , l X ) and ( Y , l Y ). We define their Cartesianproduct by ( X , l X ) × ( Y , l Y ) := ( X × Y , l X + l Y ) , where the simplicial set X × Y has n -simplices ( X × Y ) n = X n × Y n and structuremaps ( X × Y ) ϕ = X ϕ × Y ϕ for ϕ ∈ ∆ , and where ( l X + l Y )( σ, τ ) = l X ( σ ) + l Y ( τ ) . We now establish the K¨unneth formula (Theorem 1.6) that computes persistenthomology of the Cartesian product
P H ∗ ( X × Y , l X × l Y ) when X n and Y n are finitesets for n ≥ Proof of Theorem 1.6.
Recall the functor F : R + -sSet → κ [ R + ]-sMod from (6). Thereis a canonical isomorphism F ( X × Y , l X + l Y ) ∼ = F ( X , l X ) ⊗ κ [ R + ] F ( Y , l Y ) . Then the theorem follows from the definition of persistent homology (Definition 3.3)and the K¨unneth formula in κ [ R + ]-sMod (Theorem 2.10). (cid:3) Metric spaces
The main purpose of this paper is to investigate the persistent homology of thesum metric d X + d Y on the product X × Y of metric spaces ( X, d X ) and ( Y, d Y ).In Section 4.1, we define persistent homology P H ∗ ( X ) of a metric space ( X, d X )as the persistent homology P H ∗ ( X , l X ) of a R + -filtered simplicial set ( X , l X ) (Defini-tions 3.1, 3.3) associated to ( X, d X ). See Remark 4.2 for a discussion of the equivalenceof this definition to the usual Vietoris-Rips construction.The K¨unneth theorem for R + -filtered simplicial sets (Theorem 1.6) computes P H ∗ ( X × Y , l X + l Y ), however the filtration function l X × Y associated to the summetric d X + d Y is not equal to l X + l Y ; it only satisfies the inequality (see (15)) l X × Y ≤ l X + l Y . We discuss this in Section 4.2. The inequality provides a graded module homomor-phism(8)
P H ∗ ( X, Y ) :=
P H ∗ ( X × Y , l X + l Y ) → P H ∗ ( X × Y ) which fits into a long exact sequence (Proposition 4.4). The third term in the triplethat forms the long exact sequence is denoted P H ∗ ( X, Y ).In Section 4.3 we prove the vanishing
P H n ( X, Y ) = 0 in dimensions n = 0 , , , P H n ( X × Y , l X + l Y ),we obtain our K¨unneth results for P H n ( X × Y ) in low dimensions n = 0 , , P H ∗ ( X, Y ) and the homology
P H ∗ ( X × Y ) that we are interested in by the minimumof the diameters of X and Y ; see Theorem 4.9.In Section 4.5, we use Theorem 4.5 to compute P H n of the Hamming cube { , } k for n = 0 , , k ≥
1. This involves a technical computation of the nontrivialkernel of the surjection (8) in dimension 2 for X = { , } and Y = { , } k − .4.1. Persistent homology.
We associate a R + -filtered simplicial set (Definition 3.1)to every metric space, and we define the persistent homology of the metric space(Definition 4.1) to be the persistent homology of the associated filtered simplicial set(Definition 3.3). This is a simplicial set version of the usual Vietoris-Rips construction;see Remark 4.2 for further discussion.Let M et denote the category with metric spaces (
X, d X ) as objects and with mor-phisms the maps f : ( X, d X ) → ( Y, d Y ) that are distance non-increasing, i.e.(9) d Y ( f ( a ) , f ( b )) ≤ d X ( a, b ) for all a, b ∈ X. Recall from Section 3 the category R + -sSet of R + -filtered simplicial sets.We now describe a functor M et → R + -sSet( X, d X ) ( X , l X )and define persistent homology of the metric space ( X, d X ) (Definition 4.1) to be thepersistent homology of ( X , l X ).Let ( X, d X ) ∈ M et . There is a simplicial set X with n -simplices X n = { ( x , . . . , x n ) | x i ∈ X for all 0 ≤ i ≤ n } consisting of all ordered ( n + 1)-tuples of points in X . For a morphism ϕ : [ m ] → [ n ]in ∆ , the map X ϕ : X n → X m is defined on a n -simplex σ = ( x , . . . , x n ) ∈ X n by X ϕ ( x , . . . , x n ) = ( x ϕ (0) , . . . , x ϕ ( m ) ) . Note that, for the maps X ϕ to be well-defined, it is essential that the simplices σ ∈ X n are taken to be ordered tuples.There is also a max-length map l X : X n → R + (10) ( x , . . . , x n ) max { d X ( x i , x j ) | ≤ i, j ≤ n } given by the maximum of the pairwise distances between points in a simplex. Weobserve l X ( X ϕ ( σ )) ≤ l X ( σ ) holds since the points in X ϕ ( σ ) are a subset of the pointsin σ . Hence, the pair ( X , l X ) is a R + -filtered simplicial set. This defines the functor onobjects ( X, d X ) ( X , l X ). For a morphism f : ( X, d X ) → ( Y, d Y ) in M et there is anobvious induced map on simplicial sets f : X → Y that satisfies l Y ( f ( σ )) ≤ l X ( σ ) by ERSISTENT HOMOLOGY OF THE SUM METRIC 15 the distance non-increasing property (9) of f . So, f is indeed a morphism in R + -sSet.This completes the definition of the functor M et → R + -sSet.We now define the persistent homology of a metric space ( X, d X ). Definition 4.1.
Given a metric space ( X, d X ) , its persistent chain complex P C ∗ ( X ) is the persistent chain complex of the associated R + -filtered simplicial set ( X , l X ) (see Definition 3.3), i.e. P C ∗ ( X ) := P C ∗ ( X , l X ) . The persistent homology of ( X, d X ) is the homology P H ∗ ( X ) := P H ∗ ( X , l X ) ofthis chain complex.For t ∈ R + , the time- t persistent homology P H ( t ) ∗ ( X ) is the homogeneousdegree t part of P H ∗ ( X ) . Remark 4.2. (Vietoris-Rips)
The classical definition of persistent homology of ametric space ( X, d X ) uses the Vietoris-Rips construction, which we now recall. For t ∈ R + , the time- t Vietoris-Rips complex of X is the abstract simplicial complex V ( X, t ) with n -simplices given by all subsets { x , . . . , x n } ⊂ X satisfying d X ( x i , x j ) ≤ t for all ≤ i, j ≤ n . For s ≤ t there is an inclusion V ( X, s ) ֒ → V ( X, t ) whichinduces a map on homology H ∗ ( V ( X, s )) → H ∗ ( V ( X, t )) . The homology of Vietoris-Rips at all times t ∈ R + together with the induced maps of the inclusions for s ≤ t forms the persistent homology of X .Letting ( X , l X ) denote the R + -filtered simplicial set associated to ( X, d X ) , for all t ∈ R + there is a simplicial set X ( t ) := { simplices σ in X | l X ( σ ) ≤ t } . There is a homotopy equivalence V ( X, t ) → X ( t ) . The homogeneous degree- t persis-tent homology of ( X, d X ) is the homology of X ( t ) . In conclusion, for all t ∈ R + thereare isomorphisms of κ -vector spaces P H ( t ) ∗ ( X ) ∼ = H ∗ ( X ( t ) ) ∼ = H ∗ ( V ( X, t )) which are natural in t , i.e. the map H ∗ ( V ( X, s )) → H ∗ ( V ( X, t )) for s ≤ t is identifiedwith the map P H ( s ) ∗ ( X ) → P H ( t ) ∗ ( X ) given by multiplication by T t − s ∈ κ [ R + ] . A long exact sequence for the sum metric d X + d Y . Consider metric spaces X and Y with metrics d X and d Y . Ultimately, we are interested in a K¨unneth formulafor P H ∗ ( X × Y ) in terms of P H ∗ ( X ) and P H ∗ ( Y ), where the product space X × Y is equipped with the sum metric d X × Y = d X + d Y .In this section, we describe a long exact sequence in Proposition 4.4 that relates thepersistent homology of the product metric space ( X × Y, d X + d Y ) and the persistenthomology(11) P H ∗ ( X, Y ) :=
P H ∗ ( X × Y , l X + l Y )of the product filtered simplicial set X × Y with filtration function l X + l Y , where l X and l Y are the max-length maps on the simplicial sets X and Y , respectively; seeSection 4.1. This is useful because the K¨unneth theorem for filtered simplicial sets(Theorem 1.6) provides the following formula for P H ∗ ( X, Y ) in terms of
P H ∗ ( X ) and P H ∗ ( Y ). Proposition 4.3.
Given finite metric spaces ( X, d X ) and ( Y, d Y ) , for n ≥ there isa short exact sequence → M i + j = n P H i ( X ) ⊗ κ [ R + ] P H j ( Y ) → P H n ( X, Y )(12) → M i + j = n − T or ( P H i ( X ) , P H j ( Y )) → which is natural with respect to distance non-increasing maps ( X, d X ) → ( X ′ , d X ′ ) and ( Y, d Y ) → ( Y ′ , d Y ′ ) . Moreover, the sequence splits, but not naturally.Proof. By definition of persistent homology of a metric space (Definition 4.1) we have
P H ∗ ( X ) = P H ∗ ( X , l X ) and P H ∗ ( Y ) = P H ∗ ( Y , l Y ). Hence the proposition followsimmediately from Theorem 1.6 and the definition (11). (cid:3) We now describe the long exact sequence which relates
P H ∗ ( X × Y ) to P H ∗ ( X, Y ).By definition, we have the chain complexes
P C n ( X, Y ) :=
P C n ( X × Y , l X + l Y )(13) = M ( σ,τ ) ∈ X n × Y n Γ l X ( σ )+ l Y ( τ ) κ [ R + ] · h σ, τ i and(14) P C n ( X × Y ) = M ( σ,τ ) ∈ X n × Y n Γ l X × Y ( σ,τ ) κ [ R + ] · h σ, τ i . Given simplices σ = ( x , . . . , x n ) ∈ X n and τ = ( y , . . . , y n ) ∈ Y n , we claim that themax-length l X × Y ( σ, τ ) of the n -simplex ( σ, τ ) in X × Y must be less than or equalto the sum l X ( σ ) + l Y ( τ ). Indeed, for some 0 ≤ i ≤ j ≤ n , the max-length of ( σ, τ )is equal to l X × Y ( σ, τ ) = d X × Y (( x i , y i ) , ( x j , y j )) = d X ( x i , x j ) + d Y ( y i , y j ). Then since d X ( x i , x j ) ≤ l X ( σ ) and d Y ( y i , y j ) ≤ l Y ( τ ), the claim follows:(15) l X × Y ( σ, τ ) ≤ l X ( σ ) + l Y ( τ ) for all ( σ, τ ) ∈ X n × Y n . So, due to the degree shifts Γ in (13) and (14), for n ≥ R + -graded κ [ R + ]-linear embedding ι n : P C n ( X, Y ) ֒ → P C n ( X × Y )(16) h σ, τ i 7→ T l X ( σ )+ l Y ( τ ) − l X × Y ( σ,τ ) · h σ, τ i . Together these form a chain map ι : P C ∗ ( X, Y ) ֒ → P C ∗ ( X × Y ) and hence inducemaps on homology(17) ι n ∗ : P H n ( X, Y ) → P H n ( X × Y ) . Consider the relative chain complex
P C n ( X, Y ) :=
P C n ( X × Y ) /ι n ( P C n ( X, Y ))(18) = M ( σ,τ ) ∈ X n × Y n Γ l X × Y ( σ,τ ) κ [ R + ] / ( T l X ( σ )+ l Y ( τ ) − l X × Y ( σ,τ ) ) · h σ, τ i with homology P H ∗ ( X, Y ) . ERSISTENT HOMOLOGY OF THE SUM METRIC 17
Let q n : P C n ( X × Y ) → P C n ( X, Y ) denote the quotient map implicit in the defini-tion (18) and q n ∗ the induced map on homology. There is a connecting homomorphism δ n : P H n ( X, Y ) → P H n − ( X, Y ) defined in the usual way, which we now recall. Con-sider a chain α ∈ P C n ( X × Y ) that represents a cycle in P C n ( X, Y ) and hence ahomology class [ α ] ∈ P H n ( X, Y ) . This means that d ( α ) ∈ ι n − ( P C n − ( X, Y )) . Thendefine δ n ([ α ]) := [ ι − n − ( d ( α ))] ∈ P H n − ( X, Y ) . It is routine to check that δ n is inde-pendent of the choice of representative α of the class [ α ], and that it fits into thefollowing long exact sequence. Moreover, this sequence is natural in the arguments X and Y . The result is the following. Proposition 4.4.
For metric spaces ( X, d X ) and ( Y, d Y ) , there is a long exact se-quence · · · δ n +1 −−→ P H n ( X, Y ) ι n ∗ −−→ P H n ( X × Y ) q n ∗ −−→ P H n ( X, Y ) δ n −→ · · · which is natural with respect to distance non-increasing maps ( X, d X ) → ( X ′ , d X ′ ) and ( Y, d Y ) → ( Y ′ , d Y ′ ) . (cid:3) The K¨unneth Theorem for d X + d Y in low dimensions. The purpose of thissection is to prove our main Theorem 4.5, which is a refined version of Theorem 1.1from the introduction.
Theorem 4.5.
Consider metric spaces ( X, d X ) and ( Y, d Y ) and the product spaceequipped with the sum metric ( X × Y, d X + d Y ) .Then, the map ι n ∗ : P H n ( X, Y ) → P H n ( X × Y ) (see (17) ) is an isomorphism for n = 0 , , and is a surjection for n = 2 .In particular, by Proposition 4.3, if X and Y are finite sets, then for n = 0 , thereis a short exact sequence → M i + j = n P H i ( X ) ⊗ κ [ R + ] P H j ( Y ) → P H n ( X × Y )(19) → M i + j = n − T or ( P H i ( X ) , P H j ( Y )) → which is natural with respect to distance non-increasing maps ( X, d X ) → ( X ′ , d X ′ ) and ( Y, d Y ) → ( Y ′ , d Y ′ ) . Moreover, this sequence splits, but not naturally. (cid:3) Proof.
In Lemma 4.6 below we prove that
P H n ( X, Y ) = 0 for n = 0 , ,
2. Hence itfollows from the long exact sequence in Proposition 4.4 that ι n ∗ is an isomorphism for n = 0 , , and a surjection for n = 2, as claimed. The short exact sequence for n = 0 , P H n ( X, Y ) with
P H n ( X × Y ) in the short exact sequence (12)from Proposition 4.3. (cid:3) The following lemma is used in the proof of Theorem 4.5 above.
Lemma 4.6.
For n = 0 , , , the homology of the chain complex (18) vanishes P H n ( X, Y ) = 0 . In particular, the map ι n ∗ : P H n ( X, Y ) → P H n ( X × Y ) (see (17) ) is an isomor-phism for n = 0 , , and is a surjection for n = 2 . Proof.
In dimensions n = 0 and n = 1, we claim that every simplex ( σ, τ ) ∈ X n × Y n has the property l X ( σ ) + l Y ( τ ) = l X × Y ( σ, τ ). In dimension 0 this is because the max-length of any 0-simplex is zero. In dimension 1 this is because the max-length is thelength of the unique edge in the simplex. Hence, by the definition (18) of the relativechain complex, we have P C n ( X, Y ) = 0 and hence
P H n ( X, Y ) = 0 for n = 0 , n = 2. Since P C ( X, Y ) = 0 , we have P H ( X, Y ) =
P C ( X, Y ) / im ( d : P C ( X, Y ) → P C ( X, Y )) . So, we must show that the differential d : P C ( X, Y ) → P C ( X, Y )is surjective. Let ( σ, τ ) ∈ X × Y and consider the corresponding generator [ h σ, τ i ] ∈ P C ( X, Y ). To show that [ h σ, τ i ] is in the image of d we must find some α ∈ P C ( X × Y ) such that d ( α ) = h σ, τ i + r for some r ∈ ι ( P C ( X, Y )) . This is possible byLemma 4.7 below. Hence d is surjective and so P H ( X, Y ) = 0.The final statement of the lemma about ι n ∗ for n = 0 , , P H n ( X, Y ) = 0 for n = 0 , , . (cid:3) The following lemma is used in the proof of Lemma 4.6 above.
Lemma 4.7.
Given a -simplex ( σ, τ ) ∈ X × Y , there exists an element α ∈ P C ( X × Y ) such that d ( α ) = h σ, τ i + r for some r ∈ ι ( P C ( X, Y )) . Proof.
Say σ = ( x , x , x ) and τ = ( y , y , y ). Up to permutations of the indicesand switching the roles of x and y , there are four possible cases:(i) x = x = x ,(ii) x = x = x and y = y ,(iii) x = x = x and y = y ,(iv) the x i are all distinct.Notice that if h σ, τ i ∈ ι ( P C ( X, Y )), then α := 0 and r := −h σ, τ i satisfies therequired conditions. We claim that Case ( i ) and Case ( ii ) are of this form.Indeed, in Case ( i ) we have l X ( σ ) = 0, and moreover l X ( σ ) + l Y ( τ ) = l Y ( τ ) = l X × Y ( σ, τ ). Hence by definition (16) of ι , we have h σ, τ i = ι ( h σ, τ i ) ∈ ι ( P C ( X, Y )),as claimed.Similarly, in Case ( ii ), observe that l X × Y ( σ, τ ) = d X × Y (( x , y ) , ( x , y )) = d X ( x , x )+ d Y ( y , y ) = l X ( σ ) + l Y ( τ ), and so h σ, τ i = ι ( h σ, τ i ) ∈ ι ( P C ( X, Y )).We now consider Case ( iii ). Consider the 3-simplices σ ′ := ( x , x , x , x ) ∈ X ,τ ′ := ( y , y , y , y ) ∈ Y . ERSISTENT HOMOLOGY OF THE SUM METRIC 19
We claim that α := h σ ′ , τ ′ i ∈ P C ( X × Y ) satisfies the required conditions. The facesof the 3-simplex ( σ ′ , τ ′ ) ∈ X × Y are f := d ( σ ′ , τ ′ ) = ( σ, τ ) ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } . From the assumption x = x , it follows that l X × Y ( σ, τ ) = l X × Y ( σ ′ , τ ′ ). Hence bydefinition of the differential d on P C ∗ ( X × Y ) we have(20) d ( h σ ′ , τ ′ i ) = h σ, τ i + X i =1 ( − i · T l X × Y ( σ,τ ) − l X × Y ( f i ) · h f i i . Observe that, since x = x , up to a permutation of the vertices Case ( ii ) applies tothe simplices f and f and Case ( i ) applies to the simplex f . Hence for i = 1 , , , we have shown above that h f i i = ι ( h f i i ) . Hence the summation term in (20) is anelement of i ( P C ( X, Y )), i.e. d ( h σ ′ , τ ′ i ) = h σ, τ i + r where r ∈ i ( P C ( X, Y )). Thiscompletes the proof of Case ( iii ).We now consider Case ( iv ). By definition of the max-length l X × Y ( σ, τ ) and thesum metric d X × Y = d X + d Y , we have in particular d X ( x , x ) + d Y ( y , y ) = d X × Y (( x , y ) , ( x , y )) ≤ l X × Y ( σ, τ ) ,d X ( x , x ) + d Y ( y , y ) = d X × Y (( x , y ) , ( x , y )) ≤ l X × Y ( σ, τ ) . It follows that either(21) d ( x , x ) + d ( y , y ) ≤ l X × Y ( σ, τ )or d ( x , x ) + d ( y , y ) ≤ l X × Y ( σ, τ )holds. Without loss of generality, assume (21) holds.Consider the 3-simplices σ ′ := ( x , x , x , x ) ∈ X ,τ ′ := ( y , y , y , y ) ∈ Y . The faces of the 3-simplex ( σ ′ , τ ′ ) ∈ X × Y are f := d ( σ ′ , τ ′ ) = ( σ, τ ) ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } ,f := d ( σ ′ , τ ′ ) = { ( x , y ) , ( x , y ) , ( x , y ) } . By (21), an examination of all edge lengths shows that l X × Y ( σ ′ , τ ′ ) = l X × Y ( σ, τ )holds. Hence by definition of the differential d on P C ∗ ( X × Y ) we have d ( h σ ′ , τ ′ i ) = h σ, τ i + X i =1 ( − i · T l X × Y ( σ,τ ) − l X × Y ( f i ) · h f i i . Observe that, up to a permutation of the vertices and interchanging the roles of x and y , the simplices f , f and f are all of type ( i ) , ( ii ) , or ( iii ). So, the lemma saysthat for i = 1 , , , there exists α i ∈ P C ( X × Y ) such that d ( α i ) = h f i i + r i for some r i ∈ ι ( P C ( X, Y )) . We claim that α := h σ ′ , τ ′ i + X i =1 ( − i +1 · T l X × Y ( σ,τ ) − l X × Y ( f i ) · α i satisfies the required conditions; indeed, d ( α ) = h σ, τ i + X i =1 (cid:20) ( − i · T l X × Y ( σ,τ ) − l X × Y ( f i ) · h f i i + ( − i +1 · T l X × Y ( σ,τ ) − l X × Y ( f i ) · d ( α i ) (cid:21) = h σ, τ i + X i =1 ( − i · T l X × Y ( σ,τ ) − l X × Y ( f i ) · ( h f i i − d ( α i ))= h σ, τ i + X i =1 ( − i · T l X × Y ( σ,τ ) − l X × Y ( f i ) · ( − r i ) . This completes the proof of Case ( iv ), and hence the lemma. (cid:3) Interleaving distance.
We bound the interleaving distance between the mod-ule
P H ∗ ( X, Y ) defined in (11), which we can compute using the split short exactsequence in Proposition 4.3, and the homology
P H ∗ ( X × Y ) of the metric space( X × Y, d X + d Y ) that we would like to compute; see Theorem 4.9. Leonid Polterovichsuggested we look for a bound of this type.Here we view κ [ R + ]-modules as persistence vector spaces ( t P H ( t ) ∗ ( X, Y )) and( t P H ( t ) ∗ ( X × Y )); see Remark 2.1. The interleaving distance is a quantitativemeasure of how far away the computable module P H ∗ ( X, Y ) is from the true persis-tent homology
P H ∗ ( X × Y ) of the product ( X × Y, d X + d Y ). The long exact sequencein Proposition 4.4 can be viewed as an algebraic measure of the difference betweenthese modules.The general definition of interleaving distance between persistence vector spaces isas follows. The interleaving distance was introduced in [3], and has since become astandard measure of distance between persistence vector spaces. Definition 4.8.
Consider persistence vector spaces V = { V t } t ∈ R + and W = { W t } t ∈ R + with structure maps l t,t ′ : V t → V t ′ and s t,t ′ : W t → W t ′ , respectively, for t ≤ t ′ .For δ ≥ , we say that V and W are δ -interleaved if there exist linear maps F t : V t → W t + δ ,G t : W t → V t + δ such that, for all t ≤ t ′ , we have l t + δ,t ′ + δ ◦ G t = G t ′ ◦ s t,t ′ , s t + δ,t ′ + δ ◦ F t = F t ′ ◦ l t,t ′ , ERSISTENT HOMOLOGY OF THE SUM METRIC 21 and G t + δ ◦ F t = l t,t +2 δ , F t + δ ◦ G t = s t,t +2 δ . The interleaving distance between V and W is the infimum over all δ ≥ suchthat V and W are δ -interleaved. Theorem 4.9.
The interleaving distance between the persistence vector spaces ( t P H ( t ) ∗ ( X, Y )) and ( t P H ( t ) ∗ ( X × Y )) is less than or equal to the minimum of thediameters min(diameter( X ) , diameter( Y )) , where the diameter is the maximum distancediameter ( X ) = max { d X ( x , x ) | x , x ∈ X } . Proof.
Let δ := min(diameter( X ) , diameter( Y )) denote the desired bound on the in-terleaving distance. We must show that the persistence vector spaces ( t P H ( t ) ∗ ( X, Y ))and ( t P H ( t ) ∗ ( X × Y )) are δ -interleaved. Denote the structure maps on these persis-tence vector spaces by l t,t ′ : P H ( t ) ∗ ( X, Y ) → P H ( t ′ ) ∗ ( X, Y ) and s t,t ′ : P H ( t ) ∗ ( X × Y ) → P H ( t ′ ) ∗ ( X × Y ) for t ≤ t ′ . Note that the structure maps l t,t ′ and s t,t ′ on homology areinduced by the inclusions of chain complexes l t,t ′ : P C ( t ) ∗ ( X, Y ) → P C ( t ′ ) ∗ ( X, Y ) and s t,t ′ : P C ( t ) ∗ ( X × Y ) → P C ( t ′ ) ∗ ( X × Y ).We claim it suffices to show that the persistence vector spaces ( t P C ( t ) ∗ ( X, Y ))and ( t P C ( t ) ∗ ( X × Y )) are δ -interleaved with respect to chain maps F ( t ) : P C ( t ) ∗ ( X, Y ) → P C ( t + δ ) ∗ ( X × Y ) G ( t ) : P C ( t ) ∗ ( X × Y ) → P C ( t + δ ) ∗ ( X, Y ) . (The structure maps on these persistence vector spaces are the inclusions l t,t ′ and s t,t ′ .) Indeed, since F ( t ) and G ( t ) are chain maps, they induce maps on homology F ( t ) : P H ( t ) ∗ ( X, Y ) → P H ( t + δ ) ∗ ( X × Y ) and G ( t ) : P H ( t ) ∗ ( X × Y ) → P H ( t + δ ) ∗ ( X, Y ),and so the required properties of a δ -interleaving G ( t + δ ) ◦ F ( t ) = l t,t +2 δ , F ( t + δ ) ◦ G ( t ) = s t,t +2 δ ,l t + δ,t ′ + δ ◦ G ( t ) = G ( t ′ ) ◦ s t,t ′ , s t + δ,t ′ + δ ◦ F ( t ) = F ( t ′ ) ◦ l t,t ′ , follow from the corresponding properties on the chain level.We now define the maps F ( t ) and G ( t ) and show that they form a δ -interleaving onthe chain level. By definition we have P C ( t ) ∗ ( X, Y ) = M l X ( σ )+ l Y ( τ ) ≤ t κ · h σ, τ i ,P C ( t ) ∗ ( X × Y ) = M l X × Y ( σ,τ ) ≤ t κ · h σ, τ i . Since we have l X × Y ≤ l X + l Y (see (15)), it is immediate that l X × Y ≤ l X + l Y + δ holds, and so we can define F ( t ) to be the natural inclusion that takes generators togenerators F ( t ) ( h σ, τ i ) = h σ, τ i . Towards defining G ( t ) , we first establish the bound l X × Y ( σ, τ ) ≥ max( l X ( σ ) , l Y ( τ )) . Given simplices σ = ( x , . . . , x n ) ∈ X n and τ = ( y , . . . , y n ) ∈ Y n , there are integers i, j ∈ { , . . . , n } such that l X ( σ ) = d X ( x i , x j ), and similarly there are i ′ , j ′ such that l Y ( τ ) = d X ( y i ′ , y j ′ ), by definition (10) of the filtration functions l X and l Y . Hence bydefinition (10) of l X × Y and the sum metric d X × Y = d X + d Y , we have l X × Y ( σ, τ ) = max { d X × Y (( x a , y a ) , ( x b , y b )) | ≤ a, b ≤ n } = max { d X ( x a , x b ) + d Y ( y a , y b ) | ≤ a, b ≤ n }≥ max { d X ( x i , x j ) , d Y ( y i ′ , y j ′ ) } = max( l X ( σ ) , l Y ( τ )) , as claimed. From this, we derive the bound l X ( σ ) + l Y ( τ ) − l X × Y ( σ, τ ) ≤ l X ( σ ) + l Y ( τ ) − max( l X ( σ ) , l Y ( τ ))= min( l X ( σ ) , l Y ( τ )) ≤ δ. Hence we have l X + l Y ≤ l X × Y + δ , and so we can define G ( t ) to be the naturalinclusion G ( t ) ( h σ, τ i ) = h σ, τ i .It is immediate that the required properties of a δ -interleaving hold, since thestructure maps l t,t ′ and s t,t ′ as well as the interleaving maps G ( t ) and F ( t ) are all thenatural inclusions of chain complexes defined on generators by h σ, τ i 7→ h σ, τ i . Thiscompletes the proof. (cid:3)
Example: The Hamming Cube I k . Let I = { , } denote the metric spaceconsisting of 2 points at distance 1. Then for any integer k ≥
1, the Cartesian product I k with the sum metric is the Hamming k -cube , consisting of all k -tuples of zeros andones with distance given by the number of coordinates in which tuples differ.The purpose of this section is to make the following computation using Theorem 4.5.In particular, we carry out a technical computation of the nontrivial kernel of thesurjection ι ∗ : P H ( X, Y ) → P H ( X × Y ) in the case X = I and Y = I k − , k ≥ Proposition 4.10.
For k ≥ , the Hamming cube I k = { , } k satisfies the following: P H ( I k ) ∼ = (0 , k − ⊕ (0 , ∞ ) ,P H ( I k ) ∼ = (1 , k · k − − (2 k − ,P H ( I k ) = 0 . Proof.
See Proposition 4.12 and Proposition 4.13. (cid:3)
We introduce the following useful terminology for a standard coordinate embedding I r ֒ → I k . Definition 4.11.
Given ≤ r ≤ k , a point ξ ∈ I k − r , and any choice of r coordinates ≤ i < i < · · · < i r ≤ k of I k , there is an isometric embedding ϕ : I r → I k , ERSISTENT HOMOLOGY OF THE SUM METRIC 23 where for x ∈ I r the r chosen coordinates of ϕ ( x ) are equal to x , i.e. x = ( x , . . . , x r ) = ( ϕ ( x ) i , . . . , ϕ ( x ) i r ) , and the other k − r coordinates of ϕ ( x ) are given by ξ . We call these maps ϕ the coordinate inclusions of I r into I k . For r ≤ k , let C ( r, k ) denote the set of coordinate inclusions I r → I k . Proposition 4.12.
For k ≥ , we have P H ( I k ) ∼ = (0 , k − ⊕ (0 , ∞ ) ,P H ( I k ) ∼ = (1 , k · k − − (2 k − . Moreover,
P H ( I k ) is generated by the collection of images of the induced maps P H ( I ) → P H ( I k ) of the coordinate inclusions I → I k . Precisely, we have P H ( I k ) = h [ ϕ ∈ C (2 ,k ) ϕ ∗ ( P H ( I )) i . Proof.
Recall from Remark 4.2 that for t ∈ R + there is an isomorphism P H ( t ) ∗ ( I k ) ∼ = H ∗ ( V ( I k , t )), where P H ( t ) ∗ ( I k ) is the homogeneous degree t persistent homology and V ( I k , t ) is the Vietoris-Rips complex of I k at time t .A simple calculation of P H ( I k ) follows from the observation that V ( I k ,
0) is asimplicial complex consisting of 2 k points and no higher dimensional simplices, and V ( I k ,
1) is connected. Alternatively,
P H ( I k ) can be computed inductively using theK¨unneth Theorem 4.5 for metric spaces applied to I k = I × I k − , together with theobservation P H ( I ) ∼ = (0 , ⊕ (0 , ∞ ) and the formulas for tensor product and T or ofbars in Proposition 2.6.We now compute
P H ( I k ). The claimed formula holds for k = 1 since P H ( I ) = 0.We now prove the formula for k >
1, assuming inductively that it holds for k − X = I and Y = I k − together with the formulasfor tensor product and T or of bars in Proposition 2.6, we compute
P H ( I k ) ∼ = ( P H ( I ) ⊗ P H ( I k − )) ⊕ T or ( P H ( I ) , P H ( I k − )) ∼ = (1 , ( k − · k − − (2 k − ⊕ (1 , k − − = (1 , k · k − − (2 k − , as claimed.It remains to prove that P H ( I k ) is generated by the images of the induced maps ofthe coordinate inclusions I → I k . From the computed formula for P H ( I k ), we seethat it is generated by homogeneous degree 1 elements. Hence it suffices to show that P H (1)1 ( I k ) is generated by the images of the induced maps P H (1)1 ( I ) → P H (1)1 ( I k ).This is the same as showing that H ∗ ( V ( I k , H ∗ ( V ( I , → H ∗ ( V ( I k , V ( I k ,
1) is easily describable. The 0-simplices are all thepoints in I k . The 1-simplices are all pairs { x, y } such that d I k ( x, y ) = 1. Thereis a canonical identification I k ∼ = F k where F = { , } is the finite field with 2elements. Under this identification, the set of 1-simplices in V ( I k ,
1) is given by allpairs { x, x + e i } for x ∈ I k and 1 ≤ i ≤ k , where e i ∈ F k is the i -th standard basis vector. We claim that there are no higher dimensional simplices in V ( I k , { x, x + e i , y , . . . , y n } ⊂ I k is a simplex in V ( I k ,
1) forsome n ≥
1. Then d I k ( y , x ) = 1 implies y = x + e j for some standard basis vector e j . If j = i then y = x + e i , which is not allowed, and if j = i then d I k ( y , x + e i ) = 2,which is not allowed in V ( I k , V ( I ,
1) is homeomorphic to S . So, to show that H ( V ( I k , H ( V ( I , → H ( V ( I k , I → I k , it suffices to show that if we glue a disk D along the induced map on Vietoris-Rips ϕ V : ∂D ∼ = V ( I , → V ( I k ,
1) for every ϕ ∈ C (2 , k ), then every loop in V ( I k ,
1) isnullhomotopic in this new space. Precisely, we form the space X ( k ) := V ( I k , ∪ [ ϕ ∈ C (2 ,k ) D , where for each ϕ ∈ C (2 , k ) we identify ∂D ∼ = V ( I ,
1) with its image ϕ V ( V ( I , ⊂ V ( I k , . It suffices to show that every loop in V ( I k ,
1) is nullhomotopic in X ( k ). This clearlyholds for k = 1 since X ( k ) = V ( I k ,
1) = [0 ,
1] is contractible.We proceed by induction. Let k > V ( I k − ,
1) isnullhomotopic in X ( k − V ( I k ,
1) that we have aninclusion V ( I k ,
1) = ( { } × V ( I k − , ∪ ([0 , × I k − ) ∪ ( { } × V ( I k − , ⊂ [0 , × V ( I k − , . Moreover, there is an inclusion[0 , × V ( I k − , ֒ → X ( k )defined in the following way. Every 1-simplex σ in V ( I k − ,
1) has boundary points { x, x + e i } for some x ∈ I k − and standard basis vector e i ∈ I k − , and so togetherwith the first coordinate of I k = I × I k − the simplex σ determines a coordinateinclusion ϕ σ : I → I k , where the 2 chosen coordinates of I k are the first one and thecoordinate corresponding to e i ∈ I k − , and the ξ ∈ I k − is given by x in the othercoordinates. Then the inclusion [0 , × V ( I k − , ֒ → X ( k ) is defined by identifying[0 , × σ with the copy of D corresponding to ϕ σ .Consider a loop S → V ( I k , , × V ( I k − ,
1) to aloop in { } × V ( I k − , I → I k that land in { } × I k − , our loop is in the space { } × X ( k − (cid:3) Proposition 4.13.
For k ≥ , we have P H ( I k ) = 0 .Proof. By direct computation, one can check that the result holds for k ≤
3. We nowproceed by induction. Let k >
P H ( I k − ) = 0 . By Theorem 4.5, themap ι ∗ : P H ( I, I k − ) → P H ( I k ) is surjective, so it suffices to show that ι ∗ = 0.Let ϕ : I → I k − be any coordinate inclusion. Note that P H n ( I ) = 0 for n > ERSISTENT HOMOLOGY OF THE SUM METRIC 25
Proposition 4.4, we have a commutative diagram
P H ( I ) P H ( I, I ) T or ( P H ( I ) , P H ( I )) P H ( I k ) P H ( I, I k − ) T or ( P H ( I ) , P H ( I k − )) . ι ∗ α id I ∗ ⊗ ϕ ∗ ι ∗ α Moreover, since
P H n ( I ) = 0 for n > P H ( I k − ) = 0,it follows from (12) that both maps labeled α are isomorphisms.Since P H ( I ) ∼ = (0 , ⊕ (0 , ∞ ), by Proposition 2.6 we have T or ( P H ( I ) , P H ( I k − )) ∼ = T or ((0 , , P H ( I k − )) . A free resolution of the bar (0 ,
1) is given by0 → (1 , ∞ ) → (0 , ∞ ) → (0 , → . Now consider the bar (1 , ,
1) term from the free resolution andtensoring with (1 ,
2) yields (by Proposition 2.6) the sequence0 → (2 , → (1 , → . Observe in particular that the map (2 , → (1 ,
2) is necessarily zero. Hence, since
P H ( I k − ) is a direct sum of bars (1 ,
2) by Proposition 4.12, we have
T or ((0 , , P H ( I k − )) = (1 , ∞ ) ⊗ P H ( I k − ) . Hence we have shown that we have an isomorphism
T or ( P H ( I ) , P H ( I k − )) ∼ = (1 , ∞ ) ⊗ P H ( I k − ) . Under this isomorphism, the map id I ∗ ⊗ ϕ ∗ in the commutative diagram above corre-sponds to the map id (1 , ∞ ) ⊗ ϕ ∗ in the following commutative diagram, where we havealso used P H ( I ) = 0.0 P H ( I, I ) (1 , ∞ ) ⊗ P H ( I ) P H ( I k ) P H ( I, I k − ) (1 , ∞ ) ⊗ P H ( I k − ) . ∼ = id (1 , ∞ ) ⊗ ϕ ∗ ι ∗ ∼ = Now, there is a generating set of
P H ( I, I k − ) corresponding under the isomorphismwith (1 , ∞ ) ⊗ P H ( I k − ) to the generating set of P H ( I k − ) described in Proposi-tion 4.12 (tensored with the generator of (1 , ∞ )), which consists of the images of theinduced maps ϕ ∗ : P H ( I ) → P H ( I k − ) of the coordinate inclusions ϕ : I → I k − .For any such coordinate inclusion ϕ , commutativity of the above diagram shows thatthe image of (1 , ∞ ) ⊗ P H ( I ) in P H ( I k ) is 0. Hence ι ∗ = 0. (cid:3) Appendix: simplicial sets
We review basic notions about simplicial sets.Let ∆ denote the simplex category . It has objects [ n ] = { , . . . , n } for n ≥ ϕ : [ m ] → [ n ] is a morphism in ∆ if it satisfies ϕ (0) ≤ ϕ (1) ≤ · · · ≤ ϕ ( m ) . A simplicial set is a contravariant functor X : ∆ → Set . Concretely, X consists ofa set X n for each n ≥ X ϕ : X n → X m for each morphism ϕ : [ m ] → [ n ] in ∆ such that the collection of these maps satisfy the usual functorialproperties.The category of simplicial sets sSet has simplicial sets as objects and naturaltransformations as morphisms. The morphisms are called simplicial maps .There are special morphisms in ∆ called the coface maps d ∆i : [ n ] → [ n + 1] j j if j < i,j j + 1 if j ≥ i for 0 ≤ i ≤ n + 1 and the codegeneracy maps s ∆i : [ n ] → [ n − j j if j ≤ i,j j − j > i. for 0 ≤ i ≤ n −
1. These generate the morphisms in ∆ in the sense that any morphismcan be written as the composition of coface and codegeneracy maps.Given X ∈ sSet , the maps X d ∆i are called face maps and the maps X s ∆i are called degeneracy maps . Often we denote X d ∆i by d i and X s ∆i by s i when the simplicial set X is implicit. References
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