A Cartan formula for the cohomology of polyhedral products and its application to the ring structure
AA CARTAN FORMULA FOR THE COHOMOLOGY OF POLYHEDRALPRODUCTS AND ITS APPLICATION TO THE RING STRUCTURE
A. BAHRI, M. BENDERSKY, F. R. COHEN, AND S. GITLER
Abstract.
We give a geometric method for determining the cohomology groups andthe product structure of a polyhedral product Z (cid:0) K ; ( X, A ) (cid:1) , under suitable freeness con-ditions or with coefficients taken in a field k . This is done by considering first the specialcase ( X i , A i ) = ( B i ∨ C i , B i ∨ E i ) for all i , where E i (cid:44) → C i is a null homotopic inclusion,and then deriving a decomposition for these polyhedral products which resembles a Car-tan formula. The result is then generalized to arbitrary Z (cid:0) K ; ( X, A ) (cid:1) . This leads to adirect computation of the Hilbert-Poincar´e series for Z (cid:0) K ; ( X, A ) (cid:1) . Other applicationsare includedThe product structure on (cid:101) H ∗ (cid:0) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) is described in terms of the additivegenerators, labelled via the Cartan decomposition. This is done by combining with theWedge Lemma, the description in [3] of the ∗ -product structure of a polyhedral productwhich is induced from the stable splittingThe description given suffices to enable explicit calculations. Contents
1. Introduction 22. The polyhedral product of wedge decomposable pairs 43. A filtration 64. The proof of Theorem 2.2 75. Cohomological wedge decomposability and the general case 86. The canonical cofibration for wedge decomposable pairs 117. The proof of Theorem 5.4 for the boundary of a simplex 128. The proof of Theorem 5.4 for general K Z ( K ; ( X, A )) 1410. Applications 1511. Product structure 1611.1. Background 1611.2. Partial diagonals, the wedge lemma and the product of links 1811.3. Product structure for wedge decomposable pairs 2111.4. The product structure for general CW pairs 2511.5. An example 30References 31
Mathematics Subject Classification.
Primary: 52B11, 55N10, 14M25, 55U10, 13F55, Secondary:14F45, 55T10.
Key words and phrases. polyhedral product, cohomology, polyhedral smash product. a r X i v : . [ m a t h . A T ] N ov OHOMOLOGY OF POLYHEDRAL PRODUCTS 2 Introduction
Polyhedral products Z (cid:0) K ; ( X, A ) (cid:1) , [1], are defined for a simplicial complex K on thevertex set [ m ] = { , , . . . , m } , and a family of pointed CW pairs( X, A ) = (cid:8) ( X i , A i ) : i = 1 , , . . . , m (cid:9) . They are natural subspaces of the Cartesian product X × X × · · · × X m , in such a waythat if K = ∆ m − , the ( m − Z (cid:0) K ; ( X, A ) (cid:1) = X × X × · · · × X m . More specifically, we consider K to be a category where the objects are the simplices of K and the morphisms d σ,τ are the inclusions σ ⊂ τ . A polyhedral product is given as thecolimit of a diagram D ( X,A ) : K → CW ∗ , where at each σ ∈ K , we set(1.1) D ( X,A ) ( σ ) = m (cid:89) i =1 W i , where W i = (cid:40) X i if i ∈ σA i if i ∈ [ m ] − σ. Here, the colimit is a union given by Z ( K ; ( X, A )) = (cid:91) σ ∈ K D ( X,A ) ( σ ) , but the full colimit structure is used heavily in the development of the elementary theory.Notice that when σ ⊂ τ then D ( X,A ) ( σ ) ⊆ D ( X,A ) ( τ ). In the case that K itself is a simplex, Z ( K ; ( X, A )) = m (cid:89) i =1 X i . Polyhedral products were formulated first for the case ( X i , A i ) = ( D , S ) by V. Buch-staber and T. Panov in [7]; they called their spaces moment-angle complexes .In a way entirely similar to that above, a related space (cid:98) Z ( K ; ( X, A )), called the poly-hedral smash product , is defined by replacing the Cartesian product everywhere above bythe smash product. That is, (cid:98) D ( X,A ) ( σ ) = m (cid:94) i =1 W i and (cid:98) Z ( K ; ( X, A )) = (cid:91) σ ∈ K (cid:98) D ( X,A ) ( σ )with (cid:98) Z ( K ; ( X, A )) ⊆ m (cid:94) i =1 X i . The polyhedral smash product is related to the polyhedral product by the stable decom-position discussed in [1] and [2]. We denote by (
X, A ) J the restricted family of CW-pairs (cid:8) ( X j , A j (cid:1) } j ∈ J , and by K J , the full subcomplex on J ⊂ [ m ]. This work was supported in part by grant 426160 from Simons Foundation. The authors are gratefulto the Fields Institute for a conducive environment during the Thematic Program: Toric Topology andPolyhedral Products. The first author acknowledges Rider University for a Spring 2020 research leave.
OHOMOLOGY OF POLYHEDRAL PRODUCTS 3
Theorem 1.1. [2, Theorem 2.10]
Let K be an abstract simplicial complex on vertices [ m ] .Given a family { ( X j , A j ) } mj =1 of pointed pairs of CW-complexes, there is a natural pointedhomotopy equivalence (1.2) H : Σ (cid:0) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) −→ Σ (cid:0) (cid:95) J ⊆ [ m ] (cid:98) Z (cid:0) K J ; ( X, A ) J (cid:1)(cid:1) In many of the most important cases, the spaces (cid:98) Z (cid:0) K J ; ( X, A ) J (cid:1)(cid:1) can be identifiedexplicitly, [2]. Moreover, it is shown by the authors in [3] that for based CW pairs( X, A ), the product structure on the cohomology of the polyhedral product has a canonicalformulation in terms of partial diagonal maps on these spaces, (reviewed in Section 11.1).Aside from the various unstable and stable splitting theorems, [1, 12, 11, 13, 14], thereis an extensive history of computations of the cohomology groups and rings of variousfamilies of polyhedral products, [5, Sections 5, 8 and 11], see also [15, 10, 6, 18, 19, 4, 8, 9].Some very early calculations of the cohomology of certain moment-angle complexes,(the case ( X i , A i ) = ( D , S ) for all i = 1 , , . . . , m ), appeared in the work of SantiagoL´opez de Medrano [15], though at that time the spaces he studied were not recognized tohave the structure of a moment-angle complex. The cohomology algebras of all moment-angle complexes was computed first by M. Franz [10] and by I. Baskakov, V. Buchstaberand T. Panov in [6].The cohomology of the polyhedral product Z (cid:0) K ; ( X, A ) (cid:1) , for ( X, A ), satisfying certainfreeness conditions, (coefficients in a field k for example), was computed using a spectralsequence by the authors in [4]. A computation using different methods by Q. Zheng canbe found in [18, 19].The description herein of the product structure of (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) is the most explicitof which we know.As announced in [5, Section 12], one goal of the current paper is to show that forcertain pairs ( U , V ), called wedge decomposable , the algebraic decomposition given by thespectral sequence calculation [4, Theorem 5 .
4] is a consequence of an underlying geometricsplitting.
Moreover, the results of this observation extend to general based CW-pairs offinite type.
This paper is partly a revised version of the authors’ unpublished preprint from 2014,which in turn originated from an earlier preprint from 2010. In addition, the results havebeen extended now to describe the product structure of the cohomology.We begin in Section 2 by defining wedge decomposable pairs ( U , V ) and deriving forthem an explicit decomposition of the polyhedral product into a wedge of much simplerspaces, (Theorem 2.2 and Corollary 2.5). In particular, this allows us to identify explicitadditive generators for H ∗ (cid:0) Z (cid:0) K ; ( U , V ) (cid:1) . The proof in Section 4 is an induction basedon a filtration of the polyhedral product which is introduced in Section 3.A notion of cohomological wedge decomposability is introduced in Section 5 and is usedin Sections 6 to 8 to extend the results described above for H ∗ (cid:0) Z (cid:0) K ; ( U , V ) (cid:1) , to thecohomology of arbitrary polyhedral products Z (cid:0) K ; ( X, A ) (cid:1) . Given a collection ( X, A ),we associate to it wedge decomposable pairs (
U , V ) such that there is an isomorphism of
OHOMOLOGY OF POLYHEDRAL PRODUCTS 4 groups θ ( U,V ) : (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( U , V )) (cid:1) −→ (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) Effectively, this result (Theorem 5.4) gives an explicit method for labelling additive gen-erators of the group H ∗ (cid:0) Z (cid:0) K ; ( X, A )); k (cid:1) , in terms of an appropriate choice of generatorsfor H ∗ ( X ) and H ∗ ( A ) , and the link structure of the simplicial complex K .Applications of the additive results comprise Sections 9 and 10.A discussion of the ring structure of H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) occupies Section 11. The mainresult is Theorem 11.13 which, for two classes θ ( U,V ) ( u ) and θ ( U,V ) ( v ) in (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) ,allows for the explicit determination of the product θ ( U,V ) ( u ) · θ ( U,V ) ( v ) ∈ (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) . (Of course, in general, this will not be the class θ ( U,V ) ( u · v ).)The discussion of ring structure begins in subsection 11.1 with a review from [3] of therelationship between the stable splitting of the polyhedral product and the cohomologycup product. The Cartan decomposition of Theorem 2.2 enables access to the wedgelemma which decomposes the polyhedral smash product for a wedge decomposable pair,into spaces which are indexed by links. We observe in subsections 11.2 and 11.3 the wayin which this leads to a concise description of cohomology product in the case of wedgedecomposable pairs, Theorem 11.10 and Corollary 11.11.Included also in subsection 11.2, is a new method for computing the ∗ -product of thelinks which index summands in the wedge lemma decomposition of the polyhedral smashproduct. This uses the polyhedral product Z (cid:0) K ; ( S , S ) (cid:1) where S (cid:39) S is defined by(11.12).Finally, in subsection 11.4, we describe the way in which the results for wedge de-composable pairs can be extended to arbitrary pairs ( X, A ). The fact that all pairs arecohomologically wedge decomposable, suffices to do product calculations, the main pre-occupation being the tracking of changes to the indexing links generated by a mixing ofterms arising from cup products in (cid:101) H ∗ ( X i ) and (cid:101) H ∗ ( A i ).2. The polyhedral product of wedge decomposable pairs
We begin with a definition.
Definition 2.1.
The special family of CW pairs (
U , V ) = ( B ∨ C, B ∨ E ) satisfying( U i , V i ) = ( B i ∨ C i , B i ∨ E i ) for all i , where E i (cid:44) → C i is a null homotopic inclusion, is called wedge decomposable .The fact that the smash product distributes over wedges of spaces, leads to the charac-terization of the smash polyhedral product in a way which resembles a Cartan formula . Theorem 2.2. (Cartan Formula) Let ( U , V ) = ( B ∨ C, B ∨ E ) be a wedge decomposablepair, then there is a homotopy equivalence (cid:98) Z (cid:0) K ; ( U , V ) (cid:1) −→ (cid:95) I ≤ [ m ] (cid:16) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1)(cid:17) OHOMOLOGY OF POLYHEDRAL PRODUCTS 5 which is natural with respect to maps of decomposable pairs. Of course, (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) = (cid:94) j ∈ [ m ] − I B j with the convention that (cid:98) Z (cid:0) K ∅ ; ( B, B ) ∅ (cid:1) , (cid:98) Z (cid:0) K ∅ ; ( C, E ) ∅ (cid:1) and (cid:98) Z (cid:0) K I ; ( ∅ , ∅ ) I (cid:1) = S . We can decompose (cid:98) Z (cid:0) K ; ( U , V ) (cid:1) further by applying (a generalization of) the WedgeLemma. We recall first the definition of a link. Definition 2.3.
For σ a simplex in a simplicial complex K , lk σ ( K ) the link of σ in K , isdefined to be the simplicial complex for which τ ∈ lk σ ( K ) if and only if τ ∪ σ ∈ K . Theorem 2.4. [1, Theorem 2 . , [20, Lemma 1.8] Let K be a simplicial complex on [ m ] and ( C, E ) a family of CW pairs satisfying E i (cid:44) → C i is null homotopic for all i then (cid:98) Z ( K ; ( C, E )) (cid:39) (cid:95) σ ∈ K | ∆( K ) <σ | ∗ (cid:98) D [ m ] C,E ( σ ) where | ∆( K ) <σ | ∼ = | lk σ ( K ) | , the realization of the link of σ in the simplicial complex K and (2.1) (cid:98) D [ m ] C,E ( σ ) = m (cid:94) j =1 W i j , with W i j = (cid:40) C i j if i j ∈ σE i j if i j ∈ [ m ] − σ. (cid:3) Applying this to the decomposition of Theorem 2.2, we get
Corollary 2.5.
There is a homotopy equivalence (cid:98) Z (cid:0) K ; ( U , V ) (cid:1) −→ (cid:95) I ≤ [ m ] (cid:16)(cid:0) (cid:95) σ ∈ K I | lk σ ( K I ) | ∗ (cid:98) D IC,E ( σ ) (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1)(cid:17) . where (cid:98) D IC,E ( σ ) is as in (2.1) with I replacing [ m ] . Combined with Theorem 1.1, this gives a complete description of the topological spaces Z (cid:0) K ; ( U , V ) (cid:1) for wedge decomposable pairs ( U , V ) (cid:1) .The case E i (cid:39) ∗ simplifies further by [2, Theorem 2 .
15] to give the next corollary.
Corollary 2.6.
For wedge decomposable pairs of the form ( B ∨ C, B ) , corresponding to E i (cid:39) ∗ for all i = 1 , , . . . , m , there are homotopy equivalences (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) (cid:39) (cid:98) Z (cid:0) K I ; ( C, ∗ ) I (cid:1) (cid:39) (cid:98) C I , and so Theorem 2.2 gives (cid:98) Z (cid:0) K ; ( B ∨ C, B ) (cid:1) (cid:39) (cid:87) I ≤ [ m ] (cid:0) (cid:98) C I ∧ (cid:98) B ([ m ] − I ) (cid:1) . (cid:3) Notice here that the Poincar´e series for the space (cid:98) Z (cid:0) K ; ( B ∨ C, B ) (cid:1) follows easily fromCorollary 2.6. OHOMOLOGY OF POLYHEDRAL PRODUCTS 6
Remark.
In comparing these observations with [4, Theorem 5 . (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) . Also, while Theorem 2.2 and Corollary 2.5 givea geometric underpinning for the cohomology calculation in [4, Theorem 5 .
4] for wedgedecomposable pairs, the geometric splitting does not require that
E, B or C have torsion-free cohomology 3. A filtration
We begin by reviewing the filtration on polyhedral products used for the spectral se-quence calculation in [4]. Following [4, Section 2], where more details can be found. Thelength-lexicographical ordering, ( shortlex ), on the faces of the ( m − m −
1] isinduced by an ordering on the vertices. This is the left lexicographical ordering on stringsof varying lengths with shorter strings taking precedence. The ordering gives a filtrationon ∆[ m −
1] by F t (∆[ m − (cid:91) s ≤ t σ s . In turn, this gives a total ordering on the simplices of a simplicial K on m vertices(3.1) σ = ∅ < σ < σ < . . . < σ t < . . . < σ s via the natural inclusion K ⊂ ∆[ m − . This is filtration preserving in the sense that F t K = K ∩ F t ∆[ m − Example 3.1.
Consider [ m ] = [3] and K = (cid:8) φ, { v } , { v } , { v } , (cid:8) { v } , { v } (cid:9) , (cid:8) { v } , { v } (cid:9)(cid:9) with the realization consisting of two edges with a common vertex. Here the length-lexicographical ordering on the two-simplex ∆[2] is φ < v < v < v < v v < v v < v v < v v v and so the induced ordering on K is φ < v < v < v < v v < v v . Remark.
Notice that if t < m , then F t K will contain ghost vertices, that is, vertices whichare in [ m ] but are not considered simplices, They do however label Cartesian productfactors in the polyhedral product.As described in [4, Section 2], this induces a natural filtration on the polyhedral product Z (cid:0) K ; ( X, A ) (cid:1) and the smash polyhedral product (cid:98) Z (cid:0) K ; ( X, A ) (cid:1) as follows: F t Z ( K ; ( X, A )) = (cid:91) k ≤ t D ( X,A ) ( σ k ) and F t (cid:98) Z ( K ; ( X, A )) = (cid:91) k ≤ t (cid:98) D X,A ( σ k ) . Notice also that the filtration satisfies(3.2) F t (cid:98) Z ( K ; ( X, A )) = (cid:98) Z ( F t K ; ( X, A )) . OHOMOLOGY OF POLYHEDRAL PRODUCTS 7 The proof of Theorem 2.2
Let the family of CW pairs (
U , V ) be wedge decomposable as in Definition 2.1. Webegin by checking that Theorem 2.2 holds for F (cid:98) Z ( K ; ( U , V )). In this case F K consistsof the empty simplex, (the boundary of a point), and m − (cid:98) Z ( F K ; ( U , V )) = V ∧ V ∧ · · · ∧ V m (4.1) = ( B ∨ E ) ∧ ( B ∨ E ) ∧ · · · ∧ ( B m ∨ E m ) . Next, fix I = ( i , i , . . . , i k ) ⊂ [ m ] and set [ m ] − I = ( j i , j , . . . j m − k ). Then (cid:98) Z (cid:0) F K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) = ( E i ∧ E i ∧ · · · E i k ) ∧ ( B j ∧ B j ∧ · · · B j m − k ) . is the I -th wedge term in the expansion of the right hand side of (4.1). This confirmsTheorem 2.2 for t = 0.We suppose next the induction hypothesis that F t − (cid:98) Z ( K ; ( U , V )) (cid:39) (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) , with a view to verifying it for F t . The definition of the filtration gives F t (cid:98) Z ( K ; ( U , V )) = (cid:98) D U,V ( σ t ) ∪ F t − (cid:98) Z ( K ; ( U , V ))(4.2) (cid:39) (cid:98) D U,V ( σ t ) ∪ (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) . The space (cid:98) D U,V ( σ t ) is the smash product(4.3) m (cid:94) j =1 B j ∨ Y j , with Y j = (cid:40) C j if j ∈ σ t E j if j / ∈ σ t . After a shuffle of wedge factors, the space (cid:98) D U,V ( σ t ) becomes(4.4) (cid:95) I ≤ [ m ] , σ t ∈ I (cid:98) D IC,E ( σ t ) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) ∨ (cid:95) I ≤ [ m ] , σ t / ∈ I (cid:98) Z (cid:0) K I ; ( E, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) where the space (cid:98) D IC,E ( σ t ) is defined by (2.1). Remark.
Notice here the relevant fact that the number of subsets I ≤ [ m ] is the same asthe number of wedge summands in the expansion of (4.3), namely 2 m .The right-hand wedge summand in (4.4) is a subset of (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) and so, (cid:95) I ≤ [ m ] , σ t / ∈ I (cid:98) Z (cid:0) K I ; ( E, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) (cid:91) (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) = (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) . OHOMOLOGY OF POLYHEDRAL PRODUCTS 8
Finally, for each I ≤ [ m ] with σ t ∈ I , we have(4.5) (cid:98) D IC,E ( σ t ) ∪ (cid:98) Z (cid:0) F t − K I ; ( C, E ) I (cid:1) = (cid:98) Z (cid:0) F t K I ; ( C, E ) I (cid:1) . This concludes the inductive step to give(4.6) F t (cid:98) Z ( K ; ( U , V )) (cid:39) (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) F t K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( B, B ) [ m ] − I (cid:1) . It is straightforward to explicitly check the steps above in the case of F and F . Thiscompletes the proof. (cid:50) Cohomological wedge decomposability and the general case
The result of the previous section can be exploited to give information about the groups (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) over a field k for pointed, finite, path connected pairs of CW-complexes( X, A ) of finite type, which are not wedge decomposable.
Definition 5.1. A strongly free decomposition of the homology of ( X, A ) with coefficientsin a ring k , is a quadruple of k -modules ( E (cid:48) , B (cid:48) , C (cid:48) , W ) such that the long exact sequence(5.1) δ → (cid:101) H ∗ ( X/A ) (cid:96) → (cid:101) H ∗ ( X ) ι → (cid:101) H ∗ ( A ) δ → (cid:101) H ∗ +1 ( X/A ) → satisfies the condition that there exist isomorphisms(1) (cid:101) H ∗ ( A ) ∼ = B (cid:48) ⊕ E (cid:48) (2) (cid:101) H ∗ ( X ) ∼ = B (cid:48) ⊕ C (cid:48) , where B (cid:48) ι → (cid:39) B (cid:48) , ι | C (cid:48) = 0(3) (cid:101) H ∗ ( X/A ) ∼ = C (cid:48) ⊕ W , where C (cid:48) (cid:96) → (cid:39) C (cid:48) , (cid:96) | B (cid:48) = 0 , E (cid:48) δ → (cid:39) W for free graded k -modules E (cid:48) , B (cid:48) , C (cid:48) and W of finite type.A pair ( X, A ) with a strongly free decomposition is said to be strongly free . A morphismof strongly free pairs ( f (cid:96) , f ι , f δ ) : ( X, A ) → ( U, V )is a morphism of long exact sequences (5.2) · · · (cid:101) H ∗ ( X/A ) , (cid:101) H ∗ ( X ) , (cid:101) H ∗ ( A ) , (cid:101) H ∗ +1 ( X/A ) , · · ·· · · (cid:101) H ∗ ( U/V ) (cid:101) H ∗ ( U ) (cid:101) H ∗ ( V ) (cid:101) H ∗ +1 ( U/V ) · · · δ ( X,A ) (cid:96) ( X,A ) f δ ι ( X,A ) f (cid:96) δ ( X,A ) f ι l ( X,A ) f δ δ ( U,V ) (cid:96) ( U,V ) ι ( U,V ) δ ( U,V ) l ( U,V ) with all diagrams commuting, which restricts to maps of the submodules E (cid:48) , B (cid:48) , C (cid:48) , W corre-sponding to each pair. Remark.
When k is a field, the homology of ( X, A ) is always strongly free. When k = Z ,strong freeness holds if all the spaces are torsion free and of finite type.Two lemmas about null homotopic inclusions are needed next. Lemma 5.2.
Let ι : E (cid:44) → C be an inclusion of based CW complexes. then OHOMOLOGY OF POLYHEDRAL PRODUCTS 9 (1) There is a map g : cE −→ C from the reduced cone, so that the diagram belowcommutes (5.3) C ∨ cE CE E C ∨ g = h ι where h ( e ) = [( e, , that is, h includes E into the base of the cone.(2) There is a homotopy equivalence of colimits (5.4) (cid:98) Z (cid:0) ∂σ t ; ( C ∨ cE, E ) (cid:1) −→ (cid:98) Z (cid:0) ∂σ t ; ( C, E ) (cid:1) . (3) The vertical maps in (5.3) have homotopy equivalent cofibers.Proof. Since ι : E (cid:44) → C is null homotopic, the exists a homotopy G : E × I −→ C satisfying, for every e ∈ E , G (( e, ι ( e ) and G (( e, e , the base point of E , whichis also the basepoint of C . This implies the existence of an extension(5.5) E × I CcE
Gπ g where π projects E × { } to the cone point. The commutativity of (5.3) follows.Item (2) follows from the Homotopy Lemma, ([20, Lemma 1 . (cid:98) Z (cid:0) ∂σ t ; ( C ∨ cE, E ) (cid:1) and (cid:98) Z (cid:0) ∂σ t ; ( C, E ) (cid:1) .Finally, the third item is the standard homotopy invariance of cofibers, see for example[17, Theorem 2 . . (cid:3) Given a pair (
X, A ) which is strongly free, let B (cid:48) , C (cid:48) and E (cid:48) be the k –modules specifiedin items (1) and (2) of Definition 5.1 for each pair ( X, A ). Now, wedges of spheres B , C and E exist with cohomology equal to the modules B (cid:48) , C (cid:48) and E (cid:48) so that(5.6) ( U, V ) = ( B ∨ C, B ∨ E )satisfies the criterion for a wedge decomposable pair as in Definition 2.1. In particular,the inclusion ι : E → C is a null homotopic inclusion of wedges of spheres. Consider nextdiagram (5.2) for the pairs ( X, A ) and (
U, V ), the latter defined as in (5.6).
Lemma 5.3.
For a pair ( U, V ) derived from ( X, A ) in the manner above, there is anisomorphism ( f (cid:96) , f ι , f δ ) : ( X, A ) −→ ( U, V ) of strongly free pairs as in (5.1) and (5.2) .Proof. The map f l is the isomorphism (cid:101) H ∗ ( X ) ∼ = −→ B (cid:48) ⊕ C (cid:48) = (cid:101) H ∗ ( U ) and the map f ι is thecompatible isomorphism (cid:101) H ∗ ( A ) ∼ = −→ B (cid:48) ⊕ E (cid:48) = (cid:101) H ∗ ( V ), both from Definition 5.1. Since theinclusion ι : E → C is null homotopic, we apply Lemma 5.2 to write it as E (cid:44) → cE ∨ C where cE is the unreduced cone and the inclusion is onto the base of the cone. Again OHOMOLOGY OF POLYHEDRAL PRODUCTS 10 from Lemma 5.2, item (3), it follows that
C/E (cid:39) Σ E ∨ C and hence U/V (cid:39) Σ E ∨ C .This gives the isomorphism f δ : (cid:101) H ∗ ( X/A ) ∼ = −→ W ⊕ C (cid:48) = −→ (cid:101) H ∗ (Σ E ∨ C ) ∼ = −→ (cid:101) H ∗ ( U/V ) . Finally, it follows from the definitions that the diagrams (5.2) all commute. (cid:3)
Our goal is to show that under the strong freeness condition of Definition 5.1, ananalogue of Theorem 2.2 holds for pairs (
X, A ), using pairs (
U , V ) where each ( U i , V i ) hasbeen constructed from ( X i , A i ) via (5.6) above. Theorem 5.4.
Under the conditions stated above, there is an isomorphism of cohomologygroups with coefficients in a field kθ ( U,V ) : (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( U , V )) (cid:1) −→ (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) where the left hand side is determined by Corollary 2.5. (This is not necessarily an iso-morphism of modules over the Steenrod algebra and does not preserve products in general.) Corollary 5.5.
Let ( X, A ) and ( Y , B ) be two families of strongly free pairs so that thereis an isomorphism ( f (cid:96) , f ι , f δ ) of strongly free pairs ( X i , A i ) → ( Y i , B i ) for each i ∈ [ m ] asin (5.1) and (5.2) , then there is an isomorphism of groups for cohomology with coefficientsin a field k (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) −→ (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( Y , B )) (cid:1) . Proof.
For each i ∈ [ m ], the associated wedge decomposable pairs ( U i , V i ) given by (5.6)for both ( X, A ) and (
Y , B ) are the same and so the result follows from Theorem 5.4. (cid:3)
The proof of Theorem 5.4 is centered around a result of J. Grbic and S. Theriault [12],as described in [4, Section 3]. In order to keep track of ghost vertices, we introduce furthernotation. For σ = { i , i , . . . , i n +1 } , with complementary vertices { j , j , . . . , j m − n − } , set(5.7) (cid:98) A [ m ] − σ = A j ∧ A j ∧ · · · ∧ A j m − n − Theorem 5.6. [12, 4]
For general pairs ( X, A ) , the diagram below is commutative diagramof cofibrations for each t , ≤ t ≤ s . (5.8) · · · F t − (cid:98) Z ( K ; ( X, A )) F t (cid:98) Z ( K ; ( X, A )) C ( X,A ) · · ·· · · (cid:98) Z ( ∂σ t ; ( X, A )) ∧ (cid:98) A [ m ] − σ (cid:98) Zσ t ; ( X, A )) ∧ (cid:98) A [ m ] − σ C ( X,A ) · · · δ ( X,A ) t ι γ ( X,A ) t δ ( U,V ) σt g ∂σt ι γ ( U,V ) σt g σt = c t where here, σ t and ∂σ t represents the simplex σ t and the boundary of σ t respectively, consideredwithout ghost vertices, that is, as simplicial complexes on { i , i , . . . , i n +1 } , the vertices of thesimplex only. An outline of the proof of Theorem 5.4 is as follows.(1) The lower cofibration in (5.8) is analyzed geometrically in the case that (
X, A ) isa wedge decomposable pair (
U , V ). This is done in Section 6.(2) In Section 7 these results are used then to prove Theorem 5.4 for K = ∂σ t , theboundary of a simplex. OHOMOLOGY OF POLYHEDRAL PRODUCTS 11 (3) An inductive argument in Section 8 uses Diagram (5.8) to complete the proof forgeneral K .6. The canonical cofibration for wedge decomposable pairs
The three spaces in the lower cofibration of (5.8) are now analyzed in the case ofa wedge-decomposable pair (
U , V ) = ( B ∨ C, B ∨ E ). For σ t an n -simplex on vertices { i , i , . . . , i n +1 } , with no ghost vertices, the identification of the space (cid:98) Z ( σ t ; ( U , V )) isstraightforward.(6.1) (cid:98) Z ( σ t ; ( U , V )) = ( B i ∨ C i ) ∧ ( B i ∨ C i ) ∧ · · · ∧ ( B i n +1 ∨ C i n +1 ) (cid:39) C i ∧ C i ∧ · · · ∧ C i n +1 ∨ D σ t where D σ t is a wedge of smash products of spaces B i j and C i k .For general ( X, A ), the short argument in [4, Lemma 3.6] shows that the cofiber inTheorem 5.6 is given by(6.2) C ( X,A ) (cid:39) X i /A i ∧ X i /A i ∧ . . . ∧ X i n +1 /A i n +1 ∧ (cid:98) A [ m ] − σ which we write as(6.3) C ( X,A ) (cid:39) C ( X,A ) ∧ (cid:98) A [ m ] − σ Once again, for (
X, A ) = (
U , V ) = ( B ∨ C, B ∨ E ), we use Lemma 5.2 to replace theinclusion E i j (cid:44) → C i j with E i j (cid:44) → cE i j ∨ C i j where cE i j is the cone. Again, we have C i j /E i j (cid:39) Σ E i j ∨ C i j and we get the next lemma. Lemma 6.1.
The cofiber C ( U,V ) , (6.3) , for a wedge decomposable family of pairs, decom-poses homotopically into a wedge of spaces as follows. C ( U,V ) (cid:39) (Σ E i ∨ C i ) ∧ (Σ E i ∨ C i ) ∧ . . . ∧ (Σ E i n +1 ∨ C i n +1 ) (cid:39) Σ (cid:0) E i ∗ E i ∗ · · · ∗ E i n +1 (cid:1) ∨ n +1 (cid:95) j =1 C i j . ∧ Σ (cid:0) E i ∗ E i ∗ · · · ∗ (cid:98) E i j ∗ · · · ∗ E i n +1 (cid:1) ∨ n +1 (cid:95) k The polyhedral smash product (cid:98) Z (cid:0) ∂σ t ; ( C ∨ cE, E ) (cid:1) decomposes homotopi-cally into a wedge os spaces as follows. OHOMOLOGY OF POLYHEDRAL PRODUCTS 12 (cid:98) Z (cid:0) ∂σ t ; ( C, E ) (cid:1) (cid:39) (cid:98) Z (cid:0) ∂σ t ; ( C ∨ cE, E ) (cid:1) (cid:39) n +1 (cid:91) k =1 ( C i ∨ cE i ) ∧ . . . ∧ ( C i k − ∨ cE i k − ) ∧ E i k ∧ ( C i k +1 ∨ cE i k +1 ) ∧ . . . ∧ ( C i n +1 ∨ cE i n +1 ) (cid:39) E i ∗ E i ∗ · · · ∗ E i n +1 ∨ n +1 (cid:95) j =1 C i j ∧ E i ∗ E i ∗ · · · ∗ (cid:98) E i j ∗ · · · ∗ E i n +1 ∨ n +1 (cid:95) k This decomposition is the same as that given by the Wedge Lemma, [1, Theorem2 . K = ∂σ , the boundary of a simplex.Finally, comparing the two decompositions above, we arrive at the identity(6.4) C ( U,V ) (cid:39) Σ (cid:98) Z (cid:0) ∂σ t ; ( C, E ) (cid:1) ∨ (cid:0) C i ∧ C i ∧ . . . ∧ C i n +1 (cid:1) . Definition 6.3. We introduce now notational abbreviations which account for ghostvertices. Here, ( U , V ) = ( B ∨ C, B ∨ E ) and σ t , ∂σ t are as in Theorem 5.6.(1) (cid:98) Z (cid:0) σ t ; ( Y , Q ) (cid:1) = (cid:98) Z (cid:0) σ t ; ( Y , Q ) (cid:1) ∧ (cid:98) Q [ m ] − σ t for any family of pairs ( Y , Q ).(2) (cid:98) Z (cid:0) ∂σ t ; ( Y , Q ) (cid:1) = (cid:98) Z (cid:0) ∂σ t ; ( Y , Q ) (cid:1) ∧ (cid:98) Q [ m ] − σ t for any family of pairs ( Y , Q ).(3) E σ t = (cid:98) Z (cid:0) ∂σ t ; ( C, E ) (cid:1) ∧ (cid:98) V [ m ] − σ t (4) C σ t = C i ∧ C i ∧ . . . ∧ C i n +1 ∧ (cid:98) V [ m ] − σ t (5) C ( Y ,Q ) = C ( Y ,Q ) ∧ (cid:98) Q [ m ] − σ t for any family of pairs ( Y , Q ).(6) D σ t = D σ t ∧ (cid:98) V [ m ] − σ t , (see (6.1)). Theorem 6.4. For the pair ( U , V ) = ( B ∨ C, B ∨ E ) , the lower sequence correspondingto (5.8) −→ (cid:98) Z (cid:0) ∂σ t ; ( U , V ) (cid:1) i −→ (cid:98) Z (cid:0) σ t ; ( U , V ) (cid:1) γ ( U,V ) σt −−−−→ C ( U,V ) δ ( U,V ) σt −−−−→ Σ (cid:98) Z (cid:0) ∂σ t ; ( U , V ) (cid:1) i −→ splits geometrically as: (6.5) −→ E σ t ∨ D σ t i −→ C σ t ∨ D σ t γ ( U,V ) σt −−−−→ Σ E σ t ∨ C σ t δ ( U,V ) σt −−−−→ Σ E σ t ∨ Σ D σ t i −→ where the function i maps D σ t by the identity and the function γ ( U,V ) t maps C σ t by the identity. The proof of Theorem 5.4 for the boundary of a simplex Returning now to (5.8), we begin to examine the consequences of Lemma 5.3 OHOMOLOGY OF POLYHEDRAL PRODUCTS 13 Lemma 7.1. There is an isomorphism of groups (7.1) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( X, A ) (cid:1)(cid:1) ∼ = −→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( U , V ) (cid:1)(cid:1) . Proof . Applying Lemma 5.3 to Definition 6.3 part (3), and using the K¨unneth Theorem,we have (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( X, A ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( X, A ) (cid:1)(cid:1) ⊗ (cid:101) H ∗ ( (cid:98) A [ m ] − σ ) ∼ = (cid:101) H ∗ ( X i ∧ X i ∧ · · · ∧ X i n +1 ) ⊗ (cid:101) H ∗ ( A j ∧ A j ∧ · · · ∧ A j m − n − ) ∼ = (cid:101) H ∗ ( X i ) ⊗ · · · ⊗ (cid:101) H ∗ ( X i n +1 ) ⊗ (cid:101) H ∗ ( A j ) ⊗ · · · ⊗ (cid:101) H ∗ ( A j m − n − ) ∼ = (cid:101) H ∗ ( U i ) ⊗ · · · ⊗ (cid:101) H ∗ ( U i n +1 ) ⊗ (cid:101) H ∗ ( V j ) ⊗ · · · ⊗ (cid:101) H ∗ ( V j m − n − )= (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( U , V ) (cid:1)(cid:1) (cid:3) The strong freeness condition and Lemma 5.3 yield the next lemma in an analogous way. Lemma 7.2. There is an isomorphism of groups (7.2) (cid:101) H ∗ ( C ( X,A ) ) ∼ = −→ (cid:101) H ∗ ( C ( U,V ) ) , Proof . Lemma 5.3, (6.2) and the K¨unneth Theorem give (cid:101) H ∗ ( C ( X,A ) ) ∼ = (cid:101) H ∗ ( X i /A i ∧ X i /A i ∧ · · · ∧ X i n +1 /A i n +1 ) ⊗ (cid:101) H ∗ ( A j ∧ A j ∧ · · · ∧ A j m − n − ) ∼ = (cid:101) H ∗ ( X i /A i ) ⊗ · · · ⊗ (cid:101) H ∗ ( X i n +1 /A i n +1 ) ⊗ (cid:101) H ∗ ( A j ) ⊗ · · · ⊗ (cid:101) H ∗ ( A j m − n − ) ∼ = (cid:101) H ∗ ( U i /V i ) ⊗ · · · ⊗ (cid:101) H ∗ ( U i n +1 /V i n +1 ) ⊗ (cid:101) H ∗ ( V j ) ⊗ · · · ⊗ (cid:101) H ∗ ( V j m − n − ) ∼ = (cid:101) H ∗ ( C ( U,V ) ) (cid:3) The next lemma extends the isomorphism (7.1) to the boundary of a simplex. Lemma 7.3. There is an isomorphism of groups φ t : (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( U , V ) (cid:1)(cid:1) = (cid:101) H ∗ (cid:0) E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) −→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1) . Proof. Consider the ladder arising from the lower cofibration in (5.8), for both ( X, A ) and( U , A ). We have adopted the notation of (6.5) for the latter. · · · (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) σ t ; ( X, A ) (cid:1)(cid:1) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1) (cid:101) H ∗ +1 ( C ( X,A ) · · ·· · · (cid:101) H ∗ (cid:0) C σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) (cid:101) H ∗ (cid:0) E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) (cid:101) H ∗ +1 (cid:0) Σ E σ t σ t (cid:1) ⊕ (cid:101) H ∗ +1 (cid:0) C σ t (cid:1) · · · ι δ ( X,A ) σt β t ∼ = ι δ ( U,V ) σt φ t κ t ∼ = We use next the geometric splitting from (6.5) and the vertical isomorphisms to definea map φ t : (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( U , V ) (cid:1)(cid:1) = (cid:101) H ∗ (cid:0) E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) −→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1) . Theorem 6.4 gives (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( U , V ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) C σ t (cid:1) ⊕ (cid:101) H ∗ ( D σ t ) , and so, working over a field, we consider the splitting (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1)(cid:14) ( ι ◦ β t ) (cid:0) (cid:101) H ∗ ( D σ t ) (cid:1) ⊕ ( ι ◦ β t ) (cid:0) (cid:101) H ∗ ( D σ t ) (cid:1) . OHOMOLOGY OF POLYHEDRAL PRODUCTS 14 For d ∈ (cid:101) H ∗ ( D σ t ), set φ t ( d ) = ( ι ◦ β t )( d ), and for e ∈ (cid:101) H ∗ (cid:0) E σ t (cid:1) , set φ t ( e ) equal to theunique class u in (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) ∂σ t ; ( X, A ) (cid:1)(cid:1)(cid:14) ( ι ◦ β t ) (cid:0) (cid:101) H ∗ ( D σ t ) (cid:1) such that δ ( X,A ) σ t ( u ) = (cid:0) κ t ◦ δ ( U,V ) σ t (cid:1) ( e ). The diagram commutes by the construction of themap φ t . The Five-Lemma implies now that the map φ t is an isomorphism. (cid:3) The proof of Theorem 5.4 for general K We consider below a diagram of vector spaces over a field. It is constructed from thecommutative diagram (5.8) applied to the pairs ( U , V ) and ( X, A ) and incorporating theisomorphisms from Lemma 7.3. We assume by way of induction that (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) , which is true when F t − K is a simplex by (3.2) and Theorem 7.1. · · · (cid:101) H ∗ ( C ( U,V ) (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) (cid:101) H ∗ +1 ( C ( U,V ) · · ·· · · (cid:101) H ∗ (cid:0) Σ E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) C σ t (cid:1) (cid:101) H ∗ (cid:0) C σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) (cid:101) H ∗ (cid:0) E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) D σ t (cid:1) (cid:101) H ∗ +1 (cid:0) Σ E σ t (cid:1) ⊕ (cid:101) H ∗ (cid:0) C σ t (cid:1) · · ·· · · (cid:101) H ∗ ( C ( X,A ) (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) (cid:101) H ∗ +1 ( C ( X,A ) · · · γ ( U,V ) t c t ∼ = ιg σt g ∂σt δ ( U,V ) t c t ∼ = γ ( U,V ) σt ι δ ( U,V ) σt γ ( X,A ) t c t ∼ = ιg σt g ∂σt δ ( X,A ) t c t ∼ = The exactness and the commutativity of the diagram implies that we can choose isomor-phisms as follows (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) C σ t (cid:1) ⊕ L for some L (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ +1 (cid:0) Σ E σ t (cid:1) ⊕ L (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) C σ t (cid:1) ⊕ L (cid:48) for some L (cid:48) (cid:101) H ∗ (cid:0) F t − (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ +1 (cid:0) Σ E σ t (cid:1) ⊕ L (cid:48) The inductive hypothesis implies now that L ∼ = L (cid:48) and so (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( U , V ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) F t (cid:98) Z (cid:0) K ; ( X, A ) (cid:1)(cid:1) as required. This, together with the fact that result is true for a simplex and its boundary,(Section 7), completes the proof.9. The Hilbert-Poincar´e series for Z ( K ; ( X, A ))We begin by reviewing some of the elementary properties of Hilbert-Poincar´e series.Assume now that homology is taken with coefficients in a field k and all spaces arepointed, path conected with the homotopy type of CW-complexes. The Hilbert-Poincar´eseries P ( X, t ) = (cid:88) n (cid:0) dim k H n ( X ; k ) (cid:1) t n and the reduced Hilbert-Poincar´e series P ( X, t ) = − P ( X, t ) OHOMOLOGY OF POLYHEDRAL PRODUCTS 15 satisfy the following properties.(1) P ( X, t ) P ( Y, t ) = P ( X × Y ) , t ), and(2) P ( X, t ) P ( Y, t ) = P ( X ∧ Y, t ).For a pair ( X, A ) satisfying the conditions of Theorem 5.4, we have(9.1) P (cid:0) (cid:98) Z ( K ; ( X, A )) , t (cid:1) = P (cid:0) (cid:98) Z ( K ; ( U , V )) , t (cid:1) where the pair ( U, V ) is as in Definition 5.1. Next, Theorem 2.2 gives(9.2) P (cid:0) (cid:98) Z ( K ; ( U , V )) , t (cid:1) = (cid:88) I ≤ [ m ] (cid:104) P (cid:0) (cid:98) Z ( K I ; ( C, E ) I ) , t (cid:1)(cid:1) · (cid:89) j ∈ [ m ] − I P ( B i , t ) (cid:105) We apply now Corollary 2.5 to refine this further and obtain the next theorem. Theorem 9.1. The reduced Hilbert-Poincar´e series for (cid:98) Z ( K ; ( U , V )) , and hence for (cid:98) Z ( K ; ( X, A )) is given as follows, P (cid:0) (cid:98) Z ( K ; ( U , V )) , t (cid:1) = (cid:88) I ≤ [ m ] (cid:104) (cid:88) σ ∈ K I (cid:2) ( t ) P ( | lk σ ( K I ) | , t ) · P (cid:0) (cid:98) D IC,E ( σ ) (cid:1)(cid:3) · (cid:89) j ∈ [ m ] − I P ( B i , t ) (cid:105) where P (cid:0) (cid:98) D IC,E ( σ ) (cid:1) can be read off from (2.1) . Finally, Theorem 1.1 gives now the Hilbert-Poincar´e series for Z ( L ; ( U, V )) and for Z ( L ; ( X, A ), by applying (9.1) for each K = L J , J ⊆ [ m ].10. Applications Example 10.1. Consider the composite(10.1) f : C P → C P / C P ι (cid:44) −→ C P / C P where the map ι is the inclusion of the bottom two cells. Denote the mapping cylin-der of (10.1) by M f and consider the pair ( M f , C P ) for which we shall determine (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K ; ( M f , C P ) (cid:1)(cid:1) , and hence, H ∗ (cid:0) Z ( K ; ( M f , C P ) (cid:1)(cid:1) for any simplicial complex K on vertices [ m ]. Here, ( U, V ) = (cid:16) (cid:95) k =2 S k ∨ (cid:95) k =4 S k , (cid:95) k =2 S k ∨ S (cid:17) . so that B = (cid:87) k =2 S k , C = (cid:87) k =4 S k and E = S . Theorem 5.4 gives now (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K ; ( M f , C P ) (cid:1)(cid:1) ∼ = (cid:101) H ∗ (cid:0) (cid:98) Z ( K ; ( U , V )) (cid:1) . Applying Theorem 2.2, we get (cid:98) Z ( K ; ( U, V )) (cid:39) −→ (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) K I ; (cid:0) (cid:95) k =4 S k , S (cid:1)(cid:1) ∧ (cid:98) Z (cid:0) K [ m ] − I ; ( (cid:95) k =2 S k , (cid:95) k =2 S k (cid:1)(cid:1) = (cid:95) I ≤ [ m ] (cid:98) Z (cid:0) K I ; (cid:0) (cid:95) k =4 S k , S (cid:1)(cid:1) ∧ ( (cid:95) k =2 S k (cid:1) ∧| [ m ] − I | where the last term represents the ( | [ m ] − I ) | -fold smash product. Finally, Corollary 2.5determines completely each term (cid:98) Z (cid:0) K I ; (cid:0) (cid:95) k =4 S k , S (cid:1)(cid:1) OHOMOLOGY OF POLYHEDRAL PRODUCTS 16 by enumerating all the links | lk σ ( K I ) | .Theorem 2.2 applies particularly well in cases where spaces have unstable attachingmaps. Example 10.2. The homotopy equivalence S ∧ Y (cid:39) Σ( Y ) implies homotopy equivalences(10.2) Σ mq (cid:0) (cid:98) Z ( K ; ( X, A )) (cid:1) −→ (cid:98) Z (cid:0) K ; (cid:0) Σ q ( X ) , Σ q ( A ) (cid:1)(cid:1) where as usual, m is the number of vertices of K . Recall now that SO (3) ∼ = R P andconsider the pair ( X, A ) = (cid:0) SO (3) , R P (cid:1) , for which there is a well known homotopy equivalence of pairs, [16, Section 1],(10.3) (cid:0) Σ (cid:0) SO (3) (cid:1) , Σ (cid:0) R P (cid:1)(cid:1) −→ (cid:0) Σ (cid:0) R P (cid:1) ∨ Σ (S ) , Σ (cid:0) R P (cid:1)(cid:1) , which makes the pair (cid:0) SO (3) , R P (cid:1) stably wedge decomposable . Next, combining (10.2)and (10.3), we get a homotopy equivalenceΣ m (cid:0) (cid:98) Z ( K ; ( SO (3) , R P )) (cid:1) −→ (cid:98) Z (cid:0) K; (cid:0) Σ ( R P ) ∨ Σ (S ) , Σ ( R P ) (cid:1) . Finally, Theorem 5.4 allows us to conclude that (cid:98) Z ( K ; ( SO (3) , R P )) (cid:1) , and hence thepolyhedral product Z ( K ; ( SO (3) , R P )), is stably a wedge of smash products of S and R P .Similar splitting results exist for the polyhedral product whenever the spaces X and A split after finitely many suspensions. In particular, the fact that Ω S splits stably into awedge of Brown–Gitler spectra implies that the polyhedral product Z (cid:0) K ; (Ω S , ∗ ) (cid:1) splitsstably into a wedge of smash products of Brown–Gitler spectra.11. Product structure The purpose of this section is to describe the product structure of (cid:101) H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) in terms of the names of the additive generators given by Theorem 5.4, Corollary 2.5 and(11.9) below. We continue to work under the strong freeness conditions of Definition 5.1,which are satisfied, in particular, if we take coefficients in a field k .11.1. Background. We begin with a brief summary of the properties of partial diagonal maps from [3] and [4]. The main theorem of [3] asserts that the product structure onthe algebra H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) , where K is a simplicial complex on [ m ], is determinedcompletely by partial diagonals defined for P, Q subsets of [ m ], ([3, Section 2]),(11.1) (cid:98) ∆ P,QP ∪ Q : (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1) −→ (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1) inducing (cid:98) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1)(cid:1) ⊗ (cid:98) H ∗ (cid:0) (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1) −→ (cid:98) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1)(cid:1) . A sketch of the ideas from [4, Section 6] follows next. A family of pairs ( X, A ) P,QP ∪ Q isdefined by(11.2) (cid:2) ( X, A ) P,QP ∪ Q (cid:3) i = (cid:40) ( X i , A i ) if i ∈ ( P ∪ Q ) \ ( P ∩ Q )( X i ∧ X i , A i ∧ A i ) if i ∈ P ∩ Q. OHOMOLOGY OF POLYHEDRAL PRODUCTS 17 The map (cid:98) ∆ P,QP ∪ Q , factors as(11.3) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1) (cid:98) ψ P,QP ∪ Q −−−→ (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P,QP ∪ Q (cid:1) followed by(11.4) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P,QP ∪ Q (cid:1) (cid:98) S −→ (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1) where (cid:98) S is a shuffle map, originating from a natural rearrangement of smash productfactors at the diagram level, [3, Section 7], and (cid:98) ψ P,QP ∪ Q : ( X, A ) −→ ( X, A ) P,QP ∪ Q is inducedby the map of pairs(11.5) ( X i , A i ) (cid:55)−→ (cid:40) ( X i , A i ) if i ∈ ( P ∪ Q ) \ ( P ∩ Q )( X i ∧ X i , A i ∧ A i ) if i ∈ P ∩ Q. The connection between the partial diagonals and the cup product in H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) is given by the next commutative diagram of diagonals and projections, [3, Section 1].(11.6) Z ( K ; ( X, A )) Z ( K ; ( X, A )) ∧ Z ( K ; ( X, A )) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1) (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1) ∆ Z ( K ;( X,A )) (cid:98) Π P ∪ Q (cid:98) Π P ∧ (cid:98) Π Q (cid:98) ∆ P,QP ∪ Q The projection maps (cid:98) Π I are induced from the composition of the two projections:(11.7) Y [ m ] −→ Y I −→ (cid:98) Y I . This leads to the definition of the ∗ -product. Definition 11.1. Given cohomology classes u ∈ (cid:101) H p (cid:0) (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1)(cid:1) and v ∈ (cid:101) H q (cid:0) (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1)(cid:1) , the star product ∗ -product is defined by u ∗ v = ( (cid:98) ∆ P,QP ∪ Q ) ∗ ( u ⊗ v ) ∈ (cid:101) H p + q (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1)(cid:1) . The commutativity of diagram (11.6) implies that(11.8) ( (cid:98) Π P ∪ Q ) ∗ ( u ∗ v ) = ( (cid:98) Π P ) ∗ ( u ) (cid:96) ( (cid:98) Π Q ) ∗ ( v )where (cid:96) denotes the cup product in (cid:101) H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) . Finally, the ∗ -product endows (cid:77) I ⊆ [ m ] (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K I ; ( X, A ) I (cid:1)(cid:1) with a ring structure, and there is an isomorphism of rings(11.9) η : (cid:77) I ⊆ [ m ] (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K I ; ( X, A ) I (cid:1)(cid:1) −→ (cid:101) H ∗ (cid:0) Z ( K ; ( X, A )) (cid:1) induced from the additive isomorphism of Theorem 1.1, as described in [3, Section 1]. OHOMOLOGY OF POLYHEDRAL PRODUCTS 18 Partial diagonals, the wedge lemma and the product of links. We discussnow the partial diagonal maps given by (11.1) in the context of Theorem 2.4, which assertsthat if every E i (cid:44) → C i is null homotopic then for any I ⊆ [ m ],(11.10) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) (cid:39) (cid:95) σ ∈ K I | ∆(( K I ) <σ ) | ∗ (cid:98) D IC,E ( σ ) (cid:39) (cid:95) σ ∈ K I | lk σ ( K I ) | ∗ (cid:98) D IC,E ( σ )where (cid:98) D IC,E ( σ ) is as in (2.1). The next lemma checks that the partial diagonal mapsbehave as expected on the summands given by the wedge lemma. Lemma 11.2. The proof of the wedge lemma, [20, Section 4] , exhibits vertical embeddingsof homotopy colimts so that the following diagram commutes for I = J ∪ L , τ = σ ∩ J and ω = σ ∩ L , (11.11) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) (cid:98) Z (cid:0) K J ; ( C, E ) J (cid:1) ∧ (cid:98) Z (cid:0) K L ; ( C, E ) L (cid:1) | lk σ ( K I ) | ∗ (cid:98) D IC,E ( σ ) | lk τ ( K J ) | ∗ (cid:98) D JC,E ( τ ) ∧ | lk ω ( K L ) | ∗ (cid:98) D LC,E ( ω ) (cid:98) ∆ J,LI (cid:98) ξ J,LI γ ( σ ) γ ( τ ) ∧ γ ( ω ) Proof. The map (cid:98) ξ J,LI is the restriction of (cid:98) ∆ J,LI . (cid:3) In particular, (11.11) identifies the target under the partial diagonal (cid:98) ∆ J,LI , for every wedgesummand of (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) . In order to better understand the map (cid:98) ξ J,LI from (11.11)and its effect on the cohomology of the links, we need to consider a polyhedral productwhich contains all the links.Consider next the space S = D ∨ S and let(11.12) ι : S (cid:44) → D (cid:44) → S be the basepoint preserving inclusion which takes S to the endpoints of D ; this gives apair ( S , S ). Next, let K be a simplicial complex on [ m ] and apply Theorem 2.4 to thepolyhedral smash product (cid:98) Z ( K ; ( S , S )) to get(11.13) (cid:98) Z ( K ; ( S , S )) (cid:39) (cid:95) σ ∈ K | lk σ ( K ) | ∗ (cid:98) D [ m ] S , S ( σ ) (cid:39) (cid:95) σ ∈ K Σ | lk σ ( K ) | , since (cid:98) D [ m ] S , S ( σ ) (cid:39) S . For each σ ∈ K , the inclusion of each wedge summand given byTheorem 2.4 can be described as follows (11.14) | lk σ ( K ) | ∗ (cid:98) D [ m ] C,E ( σ ) = Σ | lk σ ( K ) | ∧ (cid:98) D [ m ] C,E ( σ ) ι (cid:44) −→ (cid:98) Z (cid:0) K ; ( S , S ) (cid:1) ∧ (cid:98) D [ m ] C,E ( σ ) . Combining diagram (11.6) with (11.13), we get the commutative diagram below. (11.15) Z (cid:0) K ; ( S , S ) (cid:1) Z (cid:0) K ; ( S , S ) (cid:1) ∧ Z (cid:0) K ; ( S , S ) (cid:1)(cid:98) Z (cid:0) K I ; ( S , S ) I (cid:1) (cid:98) Z (cid:0) K J ; ( S , S ) J (cid:1) ∧ (cid:98) Z (cid:0) K L ; ( S , S ) L (cid:1) Σ | lk σ ( K I ) | Σ | lk τ ( K J ) | ∧ Σ | lk ω ( K L ) | ∆ Z ( K ;( S ,S (cid:98) Π I (cid:98) Π J ∧ (cid:98) Π L (cid:98) ∆ J,LI π σ π τ ∧ π ω (cid:98) ξ J,LI where the vertical maps at the bottom are projections onto wedge summands. Thismotivates the next definition. OHOMOLOGY OF POLYHEDRAL PRODUCTS 19 Definition 11.3. Let α ∈ (cid:101) H ∗ ( | lk τ ( K J ) | ) and β ∈ (cid:101) H ∗ ( | lk ω ( K L ) | ), we write α ∗ β := ( (cid:98) ξ J,LI ) ∗ ( α τ,J ⊗ α ω,L )We need more information about (cid:101) H ∗ (cid:0) Z ( K ; ( S , S )) (cid:1) before undertaking an example. Lemma 11.4. We have the following isomorphism of cohomology groups for the simplicialcomplex K consisting of the empty simplex and m discrete points (cid:101) H ∗ (cid:0) Z ( K ; ( S , S )) (cid:1) ∼ = −→ (cid:77) I ⊂ [ m ] (cid:16) (cid:101) H ∗ (cid:0) (cid:95) | I |− S (cid:1) (cid:77) { i }∈ K I (cid:101) H ∗ (cid:0) Σ ∅ (cid:1)(cid:17) Proof. Here K = (cid:8) ∅ , { } , { } , . . . , { m } (cid:9) . Theorems 1.1 and 2.4 give (cid:101) H ∗ (cid:0) Z ( K ; ( S , S )) (cid:1) ∼ = (cid:77) I ⊂ [ m ] (cid:101) H ∗ (cid:0) (cid:101) Z ( K I ; ( S , S ) I ) (cid:1) ∼ = (cid:77) I ⊂ [ m ] (cid:16) (cid:77) σ ∈ K I (cid:101) H ∗ (cid:0) | lk σ ( K I ) | ∗ (cid:101) D I S ,S ( σ ) (cid:1)(cid:17) ∼ = (cid:77) I ⊂ [ m ] (cid:16) (cid:77) σ ∈ K I (cid:101) H ∗ (cid:0) Σ | lk σ ( K I ) | (cid:1)(cid:17) (11.16) ∼ = (cid:77) I ⊂ [ m ] (cid:16) (cid:101) H ∗ (cid:0) Σ | lk ∅ ( K I ) | (cid:1) (cid:77) { i }∈ K I (cid:101) H ∗ (cid:0) Σ | lk { i } ( K I ) | (cid:1)(cid:17) ∼ = (cid:77) I ⊂ [ m ] (cid:16) (cid:101) H ∗ (cid:0) (cid:95) | I |− S (cid:1) (cid:77) { i }∈ K I (cid:101) H ∗ (cid:0) Σ ∅ (cid:1)(cid:17) where (cid:101) D I S ,S ( σ ) (cid:39) S is as in (2.1). (cid:3) Example 11.5. We shall now analyze diagram (11.15) for the case that K = K is thesimplicial complex consisting of the empty simplex and m discrete points K = (cid:8) ∅ , { } , { } , . . . , { m } (cid:9) . Set I ⊂ [ m ], J and L subsets of I satisfying J ∪ L = I . Let σ ∈ K I be a simplex, we wishto compute (cid:101) H ∗ (cid:0) Σ | lk σ ( K I ) | (cid:1) (cid:98) ξ J,LI ) ∗ ←−−− (cid:101) H ∗ (cid:0) Σ | lk τ ( K J ) | (cid:1) (cid:79) (cid:101) H ∗ (cid:0) Σ | lk ω ( K L ) | (cid:1) for τ = σ ∩ J and ω = σ ∩ L . There are cases to consider:(1) σ = { i } ∈ K I (a) τ = σ ∩ J = { i } , ω = σ ∩ L = { i } , so lk τ ( K J ) = ∅ and lk ω ( K L ) = ∅ .(b) τ = σ ∩ J = ∅ , ω = σ ∩ L = { i } , so lk τ ( K J ) = K J and lk ω ( K L ) = ∅ .(2) σ = ∅ ∈ K I Here, τ = σ ∩ J = ∅ , ω = σ ∩ L = ∅ , so lk τ ( K J ) = K J and lk ω ( K L ) = K L .For each case, we wish to compute(11.17) (cid:101) H ∗ (cid:0) Σ | lk σ ( K I ) | (cid:1) (cid:98) ξ J,LI ) ∗ ←−−− (cid:101) H ∗ (cid:0) Σ | lk τ ( K J ) | (cid:1) (cid:79) (cid:101) H ∗ (cid:0) Σ | lk ω ( K L ) | (cid:1) . OHOMOLOGY OF POLYHEDRAL PRODUCTS 20 Case[1(a)]: In this case the bottom row of (11.15), which we wish to determine in coho-mology, is(11.18) (cid:101) H (Σ ∅ ) (cid:101) H (Σ ∅ ) (cid:78) (cid:101) H (Σ ∅ ) ∼ = k ∼ = k ⊗ k. ( (cid:98) ξ J,LI ) ∗ We denote the unit generators in (11.18) by α i,I , α i,J and α i,L respectively. Applyingcohomology to diagram (11.15), we see that( (cid:98) Π ∗ I ◦ π ∗ σ )( α i,J ∗ α i,L ) = ( (cid:98) Π ∗ J ◦ π ∗ τ )( α i,J ) (cid:94) ( (cid:98) Π ∗ L ◦ π ∗ ω )( α i,L ) . Corresponding to these classes, the top row of the diagram restricts to the cup productisomorphism(11.19) (cid:101) H ( S ) (cid:101) H ( S ) (cid:78) (cid:101) H ( S ) ∼ = k ∼ = k ⊗ k. ∆ ∗ Z ( K ;( S ,S ∼ = Now, since the map ( ι τ ) ∗ ⊗ ( ι ω ) ∗ is an isomorphism in this example, we conclude in thiscase that(11.20) α i,J ∗ α i,L = α i,I . Case[1(b)]: In this case the bottom row of (11.15) in cohomology, is(11.21) (cid:101) H (Σ ∅ ) (cid:101) H ∗ (Σ K J ) (cid:78) (cid:101) H (Σ ∅ ) ∼ = k ∼ = H ∗ (cid:0) (cid:95) | J |− S (cid:1) ⊗ k. ( (cid:98) ξ J,LI ) ∗ This time, we denote the unit generators in (11.21) by α i,I , β i,J and α i,L respectively.Applying cohomology to diagram (11.15), we see that( (cid:98) Π ∗ I ◦ π ∗ σ )( β i,J ∗ α i,L ) = ( (cid:98) Π ∗ J ◦ π ∗ τ )( β i,J ) (cid:94) ( (cid:98) Π ∗ L ◦ π ∗ ω )( α i,L )which is zero for dimensional reasons and we conclude(11.22) β i,J ∗ α i,L = 0 . Case[2]: This time the bottom row of (11.15), in cohomology, is(11.23) (cid:101) H ∗ ( K I ) (cid:101) H ∗ ( K J ) (cid:78) (cid:101) H ast ( K L ) ∼ = H ∗ (cid:0) (cid:95) | I |− S (cid:1) ∼ = H ∗ (cid:0) (cid:95) | J |− S (cid:1) ⊗ H ∗ (cid:0) (cid:95) | L |− S (cid:1) . ( (cid:98) ξ J,LI ) ∗ The unit generators in (11.23) are denoted this time by β i,I , β i,J and β i,L respectively.Applying cohomology to diagram (11.15), we see that( (cid:98) Π ∗ I ◦ π ∗ σ )( β i,J ∗ β i,L ) = ( (cid:98) Π ∗ J ◦ π ∗ τ )( β i,J ) (cid:94) ( (cid:98) Π ∗ L ◦ π ∗ ω )( α i,L )which is zero for dimensional reasons and so we conclude(11.24) β i,J ∗ β i,L = 0 . This ends the discussion of Example 11.5. OHOMOLOGY OF POLYHEDRAL PRODUCTS 21 An alternative addition to the toolkit for computing the ∗ -products of links is outlinedin the following remark. Remark 11.6. Let K be a simplicial complex on [ m ], I ⊂ [ m ] and σ ∈ K I . For a pairof subsets J and L of [ m ] satisfying I = J ∪ L , set τ = σ ∩ J and ω = σ ∩ L . Then( K I ) J = K J , ( K I ) L = K L and (cid:0) lk σ ( K ) (cid:1) I = lk σ ( K I ). There are also natural inclusions(11.25) (cid:0) lk σ ( K I ) (cid:1) J lk τ ( K J ) and (cid:0) lk σ ( K I ) (cid:1) L lk ω ( K L ) . ι τ ι ω In cases when these inclusions are equivalences of simplicial complexes, information aboutlink products may be gleaned from the commutative diagram Z ( lk σ ( K ); ( D , S )) Z ( lk σ ( K ); ( D , S )) ∧ Z ( lk σ ( K ); ( D , S )) (cid:98) Z (cid:0) ( lk σ ( K )) I ; ( D , S ) (cid:1) (cid:98) Z (cid:0) ( lk σ ( K )) J ; ( D , S ) (cid:1) ∧ (cid:98) Z (cid:0) ( lk σ ( K )) L ; ( D , S ) (cid:1) Σ | (cid:0) lk σ ( K ) (cid:1) I | Σ | (cid:0) lk σ ( K ) (cid:1) J | ∧ Σ | (cid:0) lk σ ( K ) (cid:1) L | . ∆ Z ( lkσ ( K );( D ,S (cid:98) Π I (cid:98) Π J ∧ (cid:98) Π L (cid:98) ∆ J,LI ∼ = ∼ = (cid:98) ∆ J,LI where the lower vertical isomorphisms are given by Theorem 2.4. In Example 11.5, theinclusions (11.25) are equivalences in cases [1(a)] and [2] but not in case [1(b)]. In thatcase however a dimension argument can be used to get (11.22).11.3. Product structure for wedge decomposable pairs. We consider now pairs ofthe form ( U , V ) = ( B ∨ C, B ∨ E ) where the inclusion E i (cid:44) → C i is null homotopic for all i ∈ { , , . . . , m } , as in Definition 2.1. We begin with the special case B = ∗ (a point),and compute the ∗ -product. In turn, this suffices to determine the ring structure of (cid:101) H ∗ (cid:0) Z ( K ; ( U , V )) (cid:1) = (cid:101) H ∗ (cid:0) Z ( K ; ( C, E )) (cid:1) by the ring isomorphism (11.9). The ∗ -product is given by diagram (11.11) above, basedon the decomposition of (cid:101) H ∗ (cid:0) Z ( K ; ( C, E )) (cid:1) given by Corollary 2.5.As usual we set I ⊂ [ m ], J and L subsets of I satisfying J ∪ L = I . Let σ ∈ K I be asimplex. Our goal then is to compute(11.26) (cid:101) H ∗ (cid:0) Σ | lk τ ( K J ) | (cid:1) ⊗ H ∗ (cid:0) (cid:98) D JC,E ( τ ) (cid:1) (cid:79) (cid:101) H ∗ (cid:0) Σ | lk ω ( K L ) | (cid:1) ⊗ H ∗ (cid:0) (cid:98) D LC,E ( ω ) (cid:1) (cid:98) ξ J,LI ) ∗ −−−→ (cid:101) H ∗ (cid:0) Σ | lk σ ( K I ) | (cid:1) ⊗ H ∗ (cid:0) (cid:98) D IC,E ( σ ) (cid:1) for τ = σ ∩ J and ω = σ ∩ L . Consider a class u ⊗ v ∈ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K J ; ( C, E ) J (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K L ; ( C, E ) L (cid:1)(cid:1) . in the left hand side of (11.26), it has the form(11.27) u ⊗ v = (cid:0) α ⊗ (cid:79) i ∈ τ c i ⊗ (cid:79) j ∈ J \ τ e j (cid:1) ⊗ (cid:0) β ⊗ (cid:79) k ∈ ω c k ⊗ (cid:79) l ∈ L \ ω e l (cid:1) where α , β are classes in (cid:0) (cid:101) H ∗ (Σ | lk τ K J | ) and (cid:0) (cid:101) H ∗ (Σ | lk ω K L | ) respectively. (Here thecohomology grading of all the classes has been suppressed.) Aside from classes in the OHOMOLOGY OF POLYHEDRAL PRODUCTS 22 cohomology of the links, the only cohomology classes which are multiplied are thosespecified by the diagonals (11.5) which arise on the intersection J ∩ L . That is( (cid:98) ∆ J,LJ ∪ L ) ∗ ( u ⊗ v ) = ( α ∗ β ) ⊗ (cid:79) i = k ∈ τ ∩ ω c i c k ⊗ (cid:79) i (cid:54) = k ∈ ( τ ∪ ω ) \ ( τ ∩ ω ) c i ⊗ c k ⊗ (cid:79) j = l ∈ τ (cid:48) ∩ ω (cid:48) e j e l ⊗ (cid:79) j (cid:54) = l ∈ ( τ (cid:48) ∪ ω (cid:48) ) \ ( τ (cid:48) ∩ ω (cid:48) ) e j ⊗ e l ∈ (cid:101) H ∗ (Σ | lk σ K I | ) ⊗ (cid:101) H ∗ ( (cid:98) D IC,E ( σ )) . where τ (cid:48) and ω (cid:48) denote J \ τ and L \ ω respectively. We arrive now at the next lemma. Lemma 11.7. The product structure of the ring (cid:101) H ∗ (cid:0) Z ( K ; ( C, E )) (cid:1) is determined by the ∗ -product. On two classes given as in (11.27) , it is evaluated by taking the ∗ -product of thelink classes, (cf. subsection 11.2), and the ordinary cohomology product, coordinate-wiseon the other cohomology factors of (11.26) which correspond. (cid:3) Remark. Though this formula is concise, the computation of the link product α ∗ β canbe an obstacle. It is described in more detail in [4, Section 7] from the point of view ofL. Cai’s work [8]. From the more practical perspective of subsection 11.2, diagram (11.15)relates the ∗ -product of these links to an actual cup product in (cid:101) H ∗ (cid:0) Z ( K ; ( S , S )) (cid:1) via(11.6), which can often be checked explicitly. We shall take this approach below.We consider next a full wedge decomposable pair ( U , V ) = ( B ∨ C, B ∨ E ) beginningwith the part of the partial diagonals given by the shuffle maps from (11.4).(11.28) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1) (cid:98) S −→ (cid:98) Z (cid:0) K P ; ( U , V ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1) For the pairs ( U , V ), (11.2) becomes (cid:2) ( U , V ) P,QP ∪ Q (cid:3) i = (cid:40) ( U i , V i ) = ( B i ∨ C i , B i ∨ E i ) if i ∈ P ∪ Q \ P ∩ Q (cid:0) U i ∧ U i , V i ∧ V i (cid:1) = ( (cid:98) B i ∨ (cid:98) C i , (cid:98) B i ∨ (cid:98) E i ) if i ∈ P ∩ Q. where here (cid:98) B i = B i ∧ B i (cid:98) C i = ( C i ∧ C i ) ∨ ( C i ∧ B i ) ∨ ( B i ∧ C i )(11.29) (cid:98) E i = ( E i ∧ E i ) ∨ ( E i ∧ B i ) ∨ ( B i ∧ E i ) . In particular, the pairs in ( U , V ) P,QP ∪ Q are all wedge decomposable. This motivates thenotational convention following. Definition 11.8. For any subsets S, T subsets of [ m ], we set the notation( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) S ∪ T = ( U , V ) S,TS ∪ T where ( (cid:101) B i ∨ (cid:101) C i , (cid:101) B i ∨ (cid:101) E i ) := (cid:40) ( B i ∨ C i , B i ∨ E i ) if i ∈ ( S ∪ T ) \ ( S ∩ T )( (cid:98) B i ∨ (cid:98) C i , (cid:98) B i ∨ (cid:98) E i ) if i ∈ S ∩ T. OHOMOLOGY OF POLYHEDRAL PRODUCTS 23 In that which follows in this section, we adopt additional notation following:(1) P and Q are subsets of [ m ](2) I ⊂ P ∪ Q (3) I = J ∪ L with J ⊂ P and L ⊂ Q, Notice that in this notation,(11.30) ( P ∪ Q ) \ I = (cid:0) ( P \ J ) ∪ ( Q \ L ) (cid:1) \ (cid:0) J ∩ ( Q \ L ) (cid:1) ∪ (cid:0) L ∩ ( P \ J ) (cid:1) . Generally, the meaning of the notation should become clear from the context. Figure 1 isa Venn diagram illustrating these sets, among other things.Next, we use(11.31) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1) = (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1) , and apply Theorem 2.2 to the right hand side, ( P ∪ Q here plays the role of [ m ] in thetheorem), to exhibit the shuffle map (11.28) on a wedge summand of (11.31) as (11.32) (cid:98) Z (cid:0) K I ; ( (cid:101) C, (cid:101) E ) I (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( (cid:101) B, (cid:101) B ) P ∪ Q − I (cid:1) (cid:98) S −−−→ (cid:98) Z (cid:0) K J ; ( C, E ) J (cid:1) ∧ (cid:98) Z (cid:0) K P − J ; ( B, B ) P − J (cid:1) ∧ (cid:98) Z (cid:0) K L ; ( C, E ) L (cid:1) ∧ (cid:98) Z (cid:0) K Q − L ; ( B, B ) Q − L (cid:1) where J ∪ L = I . In each of the the partial diagonal maps, the shuffle map is precededby the map (cid:98) ψ J,LJ ∪ L of (11.3) and (11.5)(11.33) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P ∪ Q (cid:1) (cid:98) ψ P,QP ∪ Q −−−→ (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1) . We apply now Theorem 2.2 to this and consider the same wedge summand to get(11.34) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P ∪ Q − I (cid:1) (cid:98) χ J,LI −−−−→ (cid:98) Z (cid:0) K I ; ( (cid:101) C, (cid:101) E ) I (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( (cid:101) B, (cid:101) B ) P ∪ Q − I (cid:1) . where (cid:98) χ J,LI is induced from (cid:98) ψ P,QP ∪ Q in (11.33) via Theorem 2.2. In order to analyze thisfurther, we need a lemma. Lemma 11.9. The diagram below commutes. (11.35) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) (cid:98) Z (cid:0) K I ; ( (cid:101) C, (cid:101) E ) I (cid:1)(cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) (cid:98) Z (cid:0) K I ; ( C, E ) J,LJ ∪ L (cid:1) (cid:98) χ J,LI (cid:12)(cid:12)(cid:12) (cid:98) Z (cid:16) KI ;( C,E ) I (cid:17)(cid:98) ψ J,LI = (cid:98) φ J,LI where the map (cid:98) φ J,LI is induced by the map of pairs (cid:40) ( C i , E i ) (cid:55)→ ( C i , E i ) if i ∈ J ∪ L \ J ∩ L, else if i ∈ J ∩ L, (cid:0) C i ∧ C i , E i ∧ E i ) (cid:44) → (cid:0) ( C i ∧ C i ) ∨ ( B i ∧ C i ) ∨ ( C i ∧ B i ) , ( E i ∧ E i ) ∨ ( B i ∧ E i ) ∨ ( B i ∧ E i ) (cid:1) the latter by the inclusion into the first wedge summands.Proof. This follows from the naturality of Theorem 2.2 for maps of wedge decomposablepairs. (cid:3) OHOMOLOGY OF POLYHEDRAL PRODUCTS 24 This lemma allow us now to replace (11.34) with the map(11.36) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P ∪ Q − I (cid:1) (cid:98) ψ J,LI ∧ (cid:98) ψ P − J,Q − LP ∪ Q − I −−−−−−−−−−−→ (cid:98) Z (cid:0) K I ; ( C, E ) J,LJ ∪ L (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P − J,Q − LP ∪ Q − I (cid:1) . where we have used(11.37) ( B, B ) P − J,Q − LP ∪ Q − I = ( (cid:101) B, (cid:101) B ) P ∪ Q − I . The next theorem describes now the product for the cohomology of a wedge decompos-able pair. Theorem 11.10. The partial diagonal map for a wedge decomposable pair ( U, V ) (cid:98) ∆ P,QP ∪ Q : (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P ∪ Q (cid:1) −→ (cid:98) Z (cid:0) K P ; ( U , V ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1) , is realized on each wedge summand given by Theorem 2.2 as (11.38) (cid:98) Z (cid:0) K I ; ( C, E ) I (cid:1) ∧ (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P ∪ Q − I (cid:1) (cid:98) ∆ J,LI ∧ (cid:98) ∆ P − J,Q − LP ∪ Q − I −−−−−−−−−−−−→ (cid:0) (cid:98) Z ( K J ; ( C, E ) J ) ∧ (cid:98) Z ( K P − J ( B, B ) P − J ) (cid:1) ∧ (cid:0) (cid:98) Z ( K L ; ( C, E ) L ) ∧ (cid:98) Z ( K Q − L ( B, B ) Q − L ) (cid:1) . Proof. The map is obtained in this form by using (11.37) to compose (11.36) with theshuffle map (11.32). (cid:3) This theorem enables now direct calculation of the ∗ -product on a wedge decompos-able pair by combining the calculation for ( C, E ) given by Lemma 11.7 with calculationfor( B, B ) which is described next.No links appear in the (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P ∪ Q − I (cid:1) wedge summand and we have have (cid:101) H ∗ (cid:0) (cid:98) Z ( K P ∪ Q − I ; ( B, B ) P ∪ Q − I ) (cid:1) ∼ = (cid:79) j ∈ P ∪ Q − I (cid:101) H ∗ ( B j ) . On this wedge factor, the partial diagonals are(11.39) (cid:98) ∆ P − J,Q − L ( P ∪ Q − I ) : (cid:98) Z (cid:0) K P ∪ Q − I ; ( B, B ) P ∪ Q − I (cid:1) −→ (cid:98) Z (cid:0) K P − J ; ( B, B ) P − J (cid:1) ∧ (cid:98) Z (cid:0) K Q − L ; ( B, B ) Q − L (cid:1) , inducing(11.40) (cid:79) j ∈ P − J (cid:101) H ∗ ( B j ) ⊗ (cid:79) k ∈ Q − L (cid:101) H ∗ ( B l ) −→ (cid:79) l ∈ P ∪ Q − I (cid:101) H ∗ ( B l )As in the case ( C, E ), classes in the same cohomology factors are multiplied; these arethe (cid:101) H ∗ ( B i ) with i ∈ ( P \ J ) ∩ ( Q \ L ), cf. (11.30). This is all summarized in the nextcorollary. Corollary 11.11. The ∗ -product associated to the ring (cid:101) H ∗ (cid:0) Z ( K ; ( U , V )) (cid:1) for a wedgedecomposable pair ( U , V ) , is determined by the smash product of partial diagonal maps (11.38) . In cohomology the products are described by Lemma 11.7 and by (11.40) . Thisdetermines the cup product structure in H ∗ (cid:0) K ; ( U , V ) (cid:1) by the description given in sub-section 11.1. OHOMOLOGY OF POLYHEDRAL PRODUCTS 25 The product structure for general CW pairs. We wish now to extend theseresults about products, from wedge decomposable pairs ( U , V ) = ( B ∨ C, B ∨ E ), to thecohomology of Z ( K ; ( X, A ) for general pairs ( X, A ). Our main tool is Theorem 5.4 whichasserts that, given ( X, A ), we can find wedge decomposable pairs ( U , V ) = ( B ∨ C, B ∨ E )so that as groups, there is an isomorphism(11.41) θ ( U,V ) : (cid:101) H ∗ (cid:0) Z ( K ; ( U , V ) (cid:1) −→ H ∗ (cid:0) Z ( K ; ( X, A ) (cid:1) . Here (11.41) is used to label the generators only, the product structure will be in termsof the product structure in the rings (cid:101) H ∗ ( X i ) and (cid:101) H ∗ ( A i ) and not in (cid:101) H ∗ ( B i ), (cid:101) H ∗ ( C i ) or (cid:101) H ∗ ( E i ).Unlike the previous example of wedge decomposable pairs, Lemma 11.9 will not hold asthe pairs ( X, A ) are not wedge decomposable in general. In this new situation, productsin (cid:101) H ∗ ( X i ) and (cid:101) H ∗ ( A i ) will mix the modules B (cid:48) i , C (cid:48) i and E (cid:48) i and so upset the links whichappear in the partial diagonal maps (11.11) and (11.39). Our task then is to keep trackof these changes.We begin by examining the diagonal maps for the spaces X i and A i and look at thediagonal maps (11.5) in terms of the notation adopted in Definition 5.1. For the longexact sequence δ → (cid:101) H ∗ ( X i ∧ X i (cid:14) A i ∧ A i ) (cid:96) → (cid:101) H ∗ ( X i ∧ X i ) ι → (cid:101) H ∗ ( A i ∧ A i ) δ → (cid:101) H ∗ +1 ( X i ∧ X i (cid:14) A i ∧ A i ) → the appropriate image, kernel and cokernel modules are given as follows.(1) (cid:101) H ∗ ( A i ∧ A i ) ∼ = ˇ B i ⊕ ˇ E i (2) (cid:101) H ∗ ( X i ∧ X i ) ∼ = ˇ B i ⊕ ˇ C i , where ˇ B i ι → (cid:39) ˇ B i , ι | ˇ C i = 0(3) (cid:101) H ∗ ( X i ∧ X i (cid:14) A i ∧ A i ) ∼ = ˇ C i ⊕ W (cid:48) i , where ˇ C i (cid:96) → (cid:39) ˇ C i , (cid:96) | ˇ B i = 0 , ˇ E i δ → (cid:39) W (cid:48) i .Comparing these to the graded k -modules we have for the pair ( X, A ) we getˇ B i = B (cid:48) i ⊗ B (cid:48) i ˇ C i = C (cid:48) i ⊗ C (cid:48) i ⊕ C (cid:48) i ⊗ B (cid:48) i ⊕ B (cid:48) i ⊗ C (cid:48) i (11.42) ˇ E i = E (cid:48) i ⊗ E (cid:48) i ⊕ E (cid:48) i ⊗ B (cid:48) i ⊕ B (cid:48) i ⊗ E (cid:48) i . Note: These cohomology modules are realized now by the cohomology of the spaces (cid:98) B i , (cid:98) C i and (cid:98) E i as defined in (11.29), and we continue to adopt the notation from Definition 11.8.Also, in order to keep the notation as simple as possible, we shall supress explicit mentionof the additive isomorphism θ ( U,V ) from (11.41), though its usage will be understoodthroughout the remainder of this section.Consider again the part of the partial diagonals given by the shuffle maps from (11.4)(11.43) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P,QP ∪ Q (cid:1) (cid:98) S −→ (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1) . Applying Theorem 5.4, the shuffle map for ( X, A ) can be described additively in coho-mology, in terms of wedge decomposable pairs, as follows(11.44) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( U , V ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1)(cid:1) (cid:98) S −→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1)(cid:1) , OHOMOLOGY OF POLYHEDRAL PRODUCTS 26 where as in (11.31) we have, (in the notation of Definition 11.8)(11.45) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1) = (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1) . Figure 1 below is a useful aid in visualizing all that follows. It displays all sets, simplicesand modules which are relevant to the discussion of the shuffle map for CW pairs andtheir cohomlogical wedge decompositions. (The numbers in square brackets label regionsof the Venn diagram non-uniquely.) Figure 1 The arrangement of modules in H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1)(cid:1) = H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1)(cid:1) Next, we apply Theorem 2.2 to the spaces (cid:98) Z (cid:0) K P ; ( U , V ) P and (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1) whichappear on the left hand side of (11.44), and then take the cohomology of each of thewedge summands resulting, to get the cohomological description of (11.43) below,(11.46) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K J ; ( C, E ) J (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P − J ; ( B, B ) P − J (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K L ; ( C, E ) L (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K Q − L ; ( B, B ) Q − L (cid:1)(cid:1) (cid:98) S ∗ −−−→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1)(cid:1) . The next lemma is now apposite. As usual, (Definition 11.8), we set I = J ∪ L . Lemma 11.12. In (11.46) the target of the shuffle map (cid:98) S ∗ is the summand (11.47) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K I ; ( (cid:101) C, (cid:101) E ) I (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q − I ; ( (cid:101) B, (cid:101) B ) P ∪ Q − I (cid:1)(cid:1) in the decomposition of (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1)(cid:1) given by Theorem 2.2. OHOMOLOGY OF POLYHEDRAL PRODUCTS 27 Remark. This is not obvious from (11.32) because it is at the cohomology level only thatwe can replace the shuffle map in which we are interested, (11.43), with one involvingwedge decomposable pairs (11.44). Proof. Consider again the equality of sets (11.30), (cf. Figure 1),( P ∪ Q ) \ I = (cid:0) ( P \ J ) ∪ ( Q \ L ) (cid:1) \ (cid:0) J ∩ ( Q \ L ) (cid:1) ∪ (cid:0) L ∩ ( P \ J ) (cid:1) . All the factors H ∗ ( B i ) on the left hand side of (11.46) satisfy i ∈ ( P \ J ) ∪ ( Q \ L ). Theonly ones which can be paired with H ∗ ( C i ) or H ∗ ( E i ) are those for which i is in thedisjoint union (cid:0) J ∩ ( Q \ L ) (cid:1) ∪ (cid:0) L ∩ ( P \ J ) (cid:1) ⊂ I . In this case, H ∗ ( C i ) ⊗ H ∗ ( B i ) ⊂ H ∗ ( (cid:98) C i ) and H ∗ ( E i ) ⊗ H ∗ ( B i ) ⊂ H ∗ ( (cid:98) E i )So, the factors of H ∗ ( B i ) from the left hand side of (11.46) which are “lost” are preciselythe ones for which i ∈ I . All other copies of H ∗ ( B i ) survive to appear in the right handtensor factor of (11.47). (cid:3) The main theorem of this section is next. Theorem 11.13. The additive isomorphism θ ( U,V ) of (11.41) suffices to determine ex-plicitly the partial diagonal map (11.1) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1)(cid:1) ( (cid:98) ∆ P,QP ∪ Q ) ∗ −−−−−→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1)(cid:1) , and hence the cup product structure in H ∗ (cid:0) K ; ( X, A ) (cid:1) , by the description given in sub-section 11.1. The rest of this section is devoted to the proof of Theorem 11.13. Consider a class x P ⊗ x Q in the left hand side of (11.44)(11.48) x P ⊗ x Q ∈ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( U , V ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1)(cid:1) . Theorem 2.2 describes this class as a sum of terms of the form u ⊗ v in (11.49) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K J ; ( C, E ) J (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P − J ; ( B, B ) P − J (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K L ; ( C, E ) L (cid:1)(cid:1) ⊗ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K Q − L ; ( B, B ) Q − L (cid:1)(cid:1) , each of which has the form below by Corollary 2.5 u = α ⊗ (cid:79) i ∈ τ c i ⊗ (cid:79) j ∈ J \ τ e j ⊗ (cid:79) j ∈ P \ J b j ∈ (cid:101) H ∗ (Σ | lk τ K J | ⊗ (cid:101) H ∗ ( (cid:98) D JC,E ( τ )) (cid:79) j ∈ P \ J (cid:101) H ∗ ( B j ) v = β ⊗ (cid:79) i ∈ ω c i ⊗ (cid:79) j ∈ L \ ω e j ⊗ (cid:79) j ∈ Q \ L b j ∈ (cid:101) H ∗ (Σ | lk ω K L | ⊗ (cid:101) H ∗ ( (cid:98) D LC,E ( ω )) (cid:79) j ∈ Q \ L (cid:101) H ∗ ( B j )It will be convenient to call α and β indexing links . Next, we apply the shuffle mapin cohomology (11.46), ignoring cohomological degree and keeping in mind (11.29) andFigure 1. This gives the description below in which cohomology elements are labelledby the regions in Figure 1 to which they belong. Recall that (11.5) restricts cohomologyproducts to the set P ∩ Q only, in particular, to Regions [1], [2], [3], [5] and [5] of Figure 1 OHOMOLOGY OF POLYHEDRAL PRODUCTS 28 only. From (11.42) we see that products on Region [4] are not supported. (Note thatsome relabelling in the names of classes is necessary to avoid ambiguities.) (cid:98) S ∗ ( u ⊗ v ) = (cid:98) S ∗ ( α ⊗ β ) ⊗ (cid:79) i ∈ [5] ( c i ⊗ ¯ c i ) ⊗ (cid:79) j ∈ [6] ( c j ⊗ b j ) ⊗ (cid:79) k ∈ [2] ( e k ⊗ ¯ e k ) ⊗ (cid:79) l ∈ [3] ( e l ⊗ b l )(11.50) ⊗ (cid:79) s ∈ [1] ( b s ⊗ ¯ b s ) ⊗ (cid:79) { t,u,v } otherwise ( c t ⊗ e u ⊗ b v )in the group(11.51) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( U , V ) P,QP ∪ Q (cid:1)(cid:1) = (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( (cid:101) B ∨ (cid:101) C, (cid:101) B ∨ (cid:101) E ) P ∪ Q (cid:1)(cid:1) . Notice that by (11.42) no products arising from Region [4] in Figure 1 can be supported.Next, we compose with the map induced by (11.3),(11.52) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P,QP ∪ Q (cid:1)(cid:1) ( (cid:98) ψ P,QP ∪ Q ) ∗ −−−−−→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1)(cid:1) to get the full partial diagonal map induced by (11.1),(11.53) (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( X, A ) P (cid:1) ∧ (cid:98) Z (cid:0) K Q ; ( X, A ) Q (cid:1)(cid:1) ( (cid:98) ∆ P,QP ∪ Q ) ∗ −−−−−→ (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ∪ Q ; ( X, A ) P ∪ Q (cid:1)(cid:1) and hence a computation of(11.54) u ∗ v = ( (cid:98) ∆ P,QP ∪ Q ) ∗ ( u ⊗ v ) = (cid:0) ( (cid:98) ψ P,QP ∪ Q ) ∗ ◦ (cid:98) S ∗ (cid:1) ( u ⊗ v )which is the part of x P ∗ x Q in the summand (11.49). The homomorphism (cid:98) ψ P,QP ∪ Q ) ∗ multi-plies the terms in (cid:98) S ∗ ( u ⊗ v ) corresponding to the marked regions. The consequences arediscussed below. Recall that the target of the map ( (cid:98) ψ P,QP ∪ Q ) ∗ , (11.52), is given by Corollary2.5, (cid:101) H ∗ (cid:0) (cid:98) Z ( K ( P ∪ Q ; ( X, A (cid:1) P ∪ Q )) ∼ = −→ (cid:101) H ∗ (cid:0) (cid:98) Z ( K P ∪ Q ; ( U , V ) P ∪ Q ) (cid:1) ∼ = −→ (cid:77) I ⊂ P ∪ Q (cid:0) (cid:77) σ ∈ K I (cid:101) H ∗ (Σ | lk σ ( K I ) | ) ⊗ (cid:101) H ∗ (cid:0) (cid:98) D IC,E ( σ )) (cid:1) (cid:79) (cid:101) H ∗ (cid:0) (cid:98) Z ( K ( P ∪ Q ) \ I ; ( B, B ) [ m ] − I ) (cid:1) . Next we analyze the monomials which can appear in ( (cid:98) ∆ P,QP ∪ Q ) ∗ ( u ⊗ v ). The properties ofthe modules B (cid:48) i , C (cid:48) i and E (cid:48) i from Definition 5.1 are used strongly below. All cup productsare either in (cid:101) H ∗ ( X i ) ∼ = B (cid:48) i ⊕ C (cid:48) i or in (cid:101) H ∗ ( A i ) ∼ = B (cid:48) i ⊕ E (cid:48) i .Below is a list of all cup products which can occur when ( (cid:98) ψ P,QP ∪ Q ) ∗ is applied to the class (cid:98) S ∗ ( u ⊗ v ) in the group (11.51):(1) For i ∈ Region [5], c i ⊗ ¯ c j (cid:55)→ c [5] i ∈ C (cid:48) i .(2) For j ∈ Region [6], c i ⊗ b j (cid:55)→ c [6] j ∈ C (cid:48) j .(3) For k ∈ Region [2], e k ⊗ ¯ e k (cid:55)→ e [2] k + b [2] k ∈ E (cid:48) k ⊕ B (cid:48) k .(4) For l ∈ Region [3], e l ⊗ b l (cid:55)→ e [3] l + b [3] l ∈ E (cid:48) l ⊕ B (cid:48) l .(5) For s ∈ Region [1], b s ⊗ ¯ b s (cid:55)→ b [1] s + c [1] s ∈ B (cid:48) s ⊕ C (cid:48) s .The next two lemmas will allow us to keep track of the links indexing the monomials. OHOMOLOGY OF POLYHEDRAL PRODUCTS 29 Lemma 11.14. Let K be a simplicial complex on vertices [ m ] , I ⊂ [ m ] \ { s } , σ ∈ K I and σ ∪ { s } ∈ K I ∪{ s } . Then there is a natural map (11.55) ρ I,s : lk σ ∪{ s } K I ∪{ s } −→ lk σ K I τ (cid:55)→ τ ∩ I Proof: Let τ ∈ lk σ ∪{ s } K I ∪{ s } , then τ ∪ ( σ ∪ { s } ) ∈ K I ∪{ s } and so τ ∪ σ ∈ K I ∪{ s } implying( τ ∩ I ) ∪ σ ∈ K I . (cid:3) Lemma 11.15. Let K be a simplicial complex on vertices [ m ] , I ⊂ [ m ] , σ ∈ K I and and l ∈ I . Then there is a natural inclusion, (11.56) ι l : lk σ K I \ { l } −→ lk σ K I . and the diagram below commutes. (11.57) lk σ K I \ { l } lk σ K I lk σ ∪{ s } K ( I ∪{ s } ) \ { l } lk σ ∪{ s } K I ∪{ s } ι l ι l ρ I \ { l } ,s ρ I, { s } Proof: Notice first that lk σ ∪{ s } K ( I ∪{ s } ) \ { l } = lk σ ∪{ s } K ( I \ { l } ) ∪{ s } . because I ∪ { s } ) \ { l } = ( I \ { l } ) ∪ { s } . Now let τ ∈ k σ ∪{ s } K ( I ∪{ s } ) \ { l } , then( ρ I,s ◦ ι l )( τ ) = ρ I,s ( τ ) = τ ∩ I. On the other hand, ( ι l ◦ ρ I \ l,s )( τ ) = ι l ( τ ∩ I ) = τ ∩ I. as well. (cid:3) We are in a position now to enumerate and describe the monomials, including theirindexing links, which arise from (11.50). Recall that (cid:0) ( (cid:98) ψ P,QP ∪ Q ) ∗ ◦ (cid:98) S ∗ (cid:1) ( α ⊗ β ) = α ∗ β isdescribed in subsection 11.2. The subscripts below are preserved from (11.50).(a) ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ e [2] k ⊗ e [3] l ⊗ b [1] s There are no changes to the indexing link α ∗ β here.(b) ρ ∗ I,s ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ e [2] k ⊗ e [3] l ⊗ c [1] s The class c [1] s in item (5) above will be zero unless the simplex ν = σ ∪ { s } exists in K I ∪{ s } . In which case, the indexing link α ∗ β must change to ρ ∗ I,s ( α ∗ β ). Here, thesimplex ν = σ ∪ { s } exists in K I ∪{ s } and so ν is a full subcomplex of K , in whichcase, Z (cid:0) ν ; ( X, A ) (cid:1) = (cid:81) i k ∈ ν X i k retracts off Z ( K ; ( X, A )) and the product is occurringin the subring H ∗ (cid:0) (cid:89) i k ∈ ν X i k (cid:1) .(c) ι ∗ l ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ e [2] k ⊗ b [3] l ⊗ b [1] s .In this case the appearance of the class b [3] l from item (4) above removes the factor (cid:101) H ∗ ( E l ) from (cid:101) H ∗ (cid:0) (cid:98) D IC,E ( σ ) (cid:1) , which was supporting the class e [3] l . This changes the set I ⊂ P ∪ Q to the set I \ { l } and so the indexing link α ∗ β changes to ι ∗ l ( α ∗ β ). OHOMOLOGY OF POLYHEDRAL PRODUCTS 30 (d) ( ι l ◦ ρ I \ l,s ) ∗ ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ e [2] k ⊗ b [3] l ⊗ c [1] s The class c [1] s in item (5) above will be zero unless the simplex ν = σ ∪ { s } existsin K I ∪{ s } . We have also the appearance of b [3] l as in item (c), necessitating a changefrom I to I \ { l } . According to Lemma 11.15, the indexing link changes from α ∗ β to( ι l ◦ ρ I \ l,s ) ∗ ( α ∗ β ).(e) ι ∗ k ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ b [2] k ⊗ e [3] l ⊗ b [1] s This time the link is altered by the appearance of b [2] k replacing a class in E (cid:48) k , so I isreplaced with I \ { k } and the link α ∗ β is changed to ι ∗ k ( α ∗ β ).(f) ( ι k ◦ ρ I \ k,s ) ∗ ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ b [2] k ⊗ e [3] l ⊗ c [1] s The class c [1] s in item (5) above will be zero unless the simplex ν = σ ∪ { s } existsin K I ∪{ s } . We have also the appearance of b [2] k as in item (e), necessitating a changefrom I to I \ { k } . According to Lemma 11.15, the indexing link changes from α ∗ β to( ι k ◦ ρ I \ k,s ) ∗ ( α ∗ β ).(g) ( ι k ◦ ι l ) ∗ ( α ∗ β ) ⊗ c [5] i ⊗ c [6] j ⊗ b [2] k ⊗ b [3] l ⊗ b [1] s This time we have the terms b [2] k and b [3] l appearing, requiring a change from I to I \ { l, k } . The indexing link changes from α ∗ β to ( ι k ◦ ι l ) ∗ ( α ∗ β ).(h) c [5] i ⊗ c [6] j ⊗ b [2] k ⊗ b [3] l ⊗ c [1] s Here, all three link changing terms b [2] k , b [3] l and c [1] s appear. In the manner above, thisalters the link from α ∗ β to ( ι l ◦ ι k ◦ ρ I \ { l,k } ,s ) ∗ ( α ∗ β ).This completes the proof of Theorem 11.13.11.5. An example. The methods of subsection 11.4 are used now to compute multiplica-tive structure in Example 10.1 for the case K = K the simplicial complex consisting oftwo discrete points, so that [ m ] = [2]. This is the polyhedral product Z ( K ; ( M f , C P ) (cid:1) ,where M f is the mapping cylinder of the map(11.58) f : C P → C P / C P ι (cid:44) −→ C P / C P and the map ι is the inclusion of the bottom two cells. Here (cid:101) H ∗ ( M f ) ∼ = k { b , b , c , c , c , c , c } and (cid:101) H ∗ ( C P ) ∼ = k { e , b , b } , where cohomological degree is denoted by a superscript . According to (Definition 5.1), wehave B (cid:48) = k { b , b } , B (cid:48) = k { b , b } C (cid:48) = k { c , c , c , c , c } , C (cid:48) = k { c , c , c , c , c } E (cid:48) = k { e } , E (cid:48) = k { e } OHOMOLOGY OF POLYHEDRAL PRODUCTS 31 where the subscript corresponds to the vertex and distinguishes the two different copies ofthe modules. Note also that in (cid:101) H ∗ ( M f ) there is the non-trivial cup product c i (cid:94) c i = c i and in (cid:101) H ∗ ( C P ) we have e i (cid:94) b i = b i .We consider the case P = Q = { , } , J = { } and L = { , } , σ = τ = ω = { } andbegin by determining the shuffle map (cid:98) S ∗ on terms in (11.49), P \ J = { } and Q \ L = ∅ , u = α , { } ⊗ c ⊗ b ∈ (cid:101) H ∗ ( | Σ lk { } ( K { } | ) ⊗ (cid:101) H ∗ ( (cid:98) D { } C,E ( { } )) ⊗ (cid:101) H ∗ ( B ) v = α , { , } ⊗ c ⊗ e ∈ (cid:101) H ∗ ( | Σ lk { } ( K { , } | ) ⊗ (cid:101) H ∗ ( (cid:98) D { , } C,E ( { } ))where here, the groups are summands of (cid:101) H ∗ (cid:0) (cid:98) Z (cid:0) K P ; ( U , V ) P (cid:1) and (cid:98) Z (cid:0) K Q ; ( U , V ) Q (cid:1)(cid:1) givenby Theorem 2.2. We apply now the shuffle map as in (11.50) to get (cid:98) S ∗ ( u ⊗ v ) = (cid:98) S ∗ ( α , { } ⊗ α , { , } ) ⊗ ( c ⊗ c ) ⊗ ( b ⊗ e )= α , { , } ⊗ ( c ⊗ c ) ⊗ ( b ⊗ e )by (11.20). 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