Kyoto University. Decompositions of polyhedral products. Kouyemon IRIYE and Daisuke KISHIMOTO

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1 yoto University yoto-math Decompositions of polyhedral products by ouyemon IRIYE and Daisuke ISHIMOTO October 2011 Department of Mathematics Faculty of Science yoto University yoto , JAPAN

2 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS OUYEMON IRIYE AND DAISUE ISHIMOTO Abstract. In [BBCG], Bahri, Bendersky, Cohen and Gitler posed a conjecture on decompositions of certain polyhedral products, supported by their own result on suspensions of polyhedral products and a result of Grbić and Theriault [GT]. We resolve this conjecture affirmatively. 1. Introduction In this paper, we resolve a conjecture of Bahri, Bendersky, Cohen and Gitler [BBCG] on decompositions of polyhedral products which generalizes a result of Grbić and Theriault [GT]. Throughout the paper, we work in the category of compactly generated weak Hausdorff spaces with nondegenerate base points and base point preserving maps. Let us first recall the definition of polyhedral products. Let be a simplicial complex on the index set [n] ={1,...,n}, and let (X,A) be a collection of pairs of spaces {(X i,a i )} n i=1. The polyhedral product or the generalized moment-angle complex of (X,A) with respect to, denoted by Z (X,A), is defined as the union of all i σ X i i σ A i for σ. Polyhedral products first appeared in a work of Porter [P] which defines higher order Whitehead products. Since then, polyhedral products have been studied in connection with topology, algebra and combinatorics, and so there are diverse results [B, BBCG, BP, DJ, DO, DS, FT, GT, N]. Among these results, we are particularly interested in decompositions of polyhedral products. Bahri, Bendersky, Cohen and Gitler [BBCG] gave decompositions of polyhedral products after a suspension. Let us illustrate a special example of their result. We set some notation. Let denote the geometric realization of a simplicial complex. A subcomplex L of is called induced whenever all vertices of a simplex σ of belong to L, σ is a simplex of L. We denote the induced subcomplex of on the vertex set I V () by I, where V () is the vertex set of. For a collection of spaces {X i } n i=1 and = J [n], let X J denote the smash product j J. For spaces X, Y, let X Y be the join of X and Y. Theorem 1.1 (Bahri, Bendersky, Cohen and Gitler [BBCG]). Let be a simplicial complex on the index set [n] and let (X,A) be a collection of NDR pairs {(X i,a i )} n i=1. If each X i is contractible, there is a homotopy equivalence ΣZ (X,A) Σ I V () I ÂI. There is also a result due to Grbić and Theriault [GT] on decompositions of certain polyhedral products without a suspension. To state their result, we define special simplicial complexes. A 1

3 2 OUYEMON IRIYE AND DAISUE ISHIMOTO simplicial complex is called shifted if there is an order on V () whenever v σ and v<w V (), (σ v) w belongs to. Theorem 1.2 (Grbić and Theriault [GT]). Let be a shifted complex on the index set [n] and let (D 2,S 1 ) be a collection of n-copies of (D 2,S 1 ). Then Z (D 2,S 1 ) has the homotopy type of a wedge of spheres. Supported by Theorem 1.1 and 1.2, Bahri, Bendersky, Cohen and Gitler [BBCG] posed the following conjecture. For a collection of spaces X = {X i } n i=1, put (CX,X)={(CX i,x i )} n i=1. Conjecture 1.3 (Bahri, Bendersky, Cohen and Gitler [BBCG]). Let be a shifted complex on the index set [n] and let X be a collection of spaces {X i } n i=1 such that each X i is path-connected. Then Z (CX,X) I X I I [n] The proof of Theorem 1.2 in [GT] heavily relies on the fact that S 1 has a classifying space, and then we must make a different approach to Conjecture 1.3. Our approach is simple and straightforward compared to that of Grbić and Theriault [GT], and we resolve Conjecture 1.3 together with a certain naturality with respect to (Theorem 5.2). As its corollary, we obtain: Theorem 1.4. Let be a shifted complex on the index set [n] and let X be a collection of spaces {X i } n i=1. Then there is a homotopy equivalence Z (CX,X) I X I. =I [n] Corollary 1.5. Let and X be as in Theorem 1.4. If ΣX i has the homotopy type of a wedge of spheres for each i, so does Z (CX,X). Proof. By pinching out the star of the maximum vertex, any shifted complex becomes a wedge of spheres. See Lemma 3.5 below. Then since each induced subcomplex of a shifted complex is also shifted, the proof is completed by Theorem Lemmas on pushouts The purpose of this section is to collect some lemmas on pushouts of topological spaces. Let us first recall two basic properties of pushouts of topological spaces, where we omit the proofs. We will use these two properties implicitly in what follows.

4 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 3 Lemma 2.1. Suppose there is a commutative diagram A E g f 1 h C G f 3 B D F f 2 H in which the top and the bottom faces are pushouts and g, h are cofibrations. If f 1,f 2,f 3 are homotopy equivalences, so is f 4. f 4 Lemma 2.2. If a commutative diagram A B is a pushout, so is C A E D B E C E D E. We consider a special case that pushouts preserve cofiber sequences. Lemma 2.3. Suppose there is a commutative diagram A 1 f B 1 g C 1 A 2 B 2 C 2 A 3 B 3 C 3 A 4 B 4 C 4 in which all three side faces are pushouts and f,g are cofibrations. If A i B i C i is a cofiber sequence for i =1, 2, 3, so is A 4 B 4 C 4. Proof. Since colimits commute with colimits, C 4 = B 4 /A 4 and the map B 4 C 4 is the projection. Then we show A 4 B 4 is a cofibration. By a result of Lillig [L], the map Q B 2 is a cofibration, where Q is a pushout of A 2 A 1 B 1. Then it follows from [H, Lemma ] that A 4 B 4 is also a cofibration.

5 4 OUYEMON IRIYE AND DAISUE ISHIMOTO For spaces X, Y, we put X Y = X Y/X. Lemma 2.4. Define Q as a pushout A (B C) i 1 CA (B C) 1 (1 ) A (B D) Q, where i : A CA is the usion. Then there is a homotopy equivalence Q B (A D) (ΣA C) which is natural with respect to A, B, C, D. Proof. Apply Lemma 2.3 to a commutative diagram A A i CA 1 (1 ) A (B D) A (B C) i 1 CA (B C), where the vertical arrows are the usions into the first factors. Then we get a cofiber sequence CA Q q Q, where Q is defined as a pushout A (B C) 1 (1 ) i 1 CA (B C) A (B D) Q. Obviously, Q =(CA B) (A D) (ΣA C) and then the composite of q : Q Q and the projection Q B (A D) (ΣA C) is the desired homotopy equivalence. 3. Topology of shifted complexes In this section, we study the topology of shifted complexes. Let be a shifted complex. If V () is a subset of [n], we assume that is shifted by the order of [n]. A shifted subcomplex of means a subcomplex of which is shifted by the order of V (). We give two important examples of shifted subcomplexes. For a simplicial complex and its vertex v, let rest (v), star (v) and link (v) be the induced subcomplex V () v, the star and the link of v in, respectively. Example 3.1. Let be a shifted complex. Any induced subcomplex of is a shifted subcomplex of. In particular, rest (v) is a shifted subcomplex of. Example 3.2. Let be a shifted complex. If v is the maximum vertex, star (v) is a shifted subcomplex of. Then link (v) is also a shifted subcomplex of since it is an induced subcomplex of star (v). Let us first consider the connected components of shifted complexes.

6 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 5 Proposition 3.3. Let be a shifted complex on the index set [n,n]={n,n +1,...,n} and let 0 be the connected component of containing the maximum vertex n. Then V ( 0 )= V (star (n)) = [n 0,n] for some n 0 [n,n] and 0 is discrete. Proof. Choose any vertex v of 0. Then v is adjacent to some vertex w of 0. If v<w, v is also adjacent to n, equivalently, v V (star (n)). If v>w, w is adjacent to n, implying v V (star (n)). Then we obtain V ( 0 ) V (star (n)). The converse implication is obvious and thus the first assertion is proved. We next prove the second assertion. If there are adjacent vertices v, w of 0 with v<w, one sees that v V (star (n)) as above, a contradiction. Then there is no edge in 0, that is, 0 is discrete. We next consider the quotient of a shifted complex by the star of the maximum vertex. We set some notation. Let be a shifted complex with the maximum vertex v. Let m() be the set of all maximal simplices of which do not contain v. We put m 0 () =m(rest (v)) m(link (v)), m 1 () =m(rest (v)) m 0 (), m 2 () =m(link (v)) m 0 (). Proposition 3.4. For a connected shifted complex, the dimension of any simplex in m 1 () is positive. Proof. By Proposition 3.3, we have V (link (v)) = V (rest (v)), implying if σ m(rest (v)) and dim σ = 0, σ belongs to m(link (v)), where v is the maximum vertex of. Let be a shifted complex with the maximum vertex v. Put = / star (v). For a shifted subcomplex L of satisfying v L, let ι,l : L be the induced map from the usion of L into. Proposition 3.5. Let be a shifted complex. Then = σ m() S dim σ and ι,l is a wedge of the identity map of S dim σ for σ m(l) m() and the constant map on other spheres, where L is a shifted subcomplex of having the common maximum vertex. Proof. Any simplex σ of satisfies σ star (v), where v is the maximum vertex of. Then the first assertion follows. The second assertion is obvious.

7 6 OUYEMON IRIYE AND DAISUE ISHIMOTO Proposition 3.6. Let be a shifted complex with the maximum vertex v. There is an identification =( S dim σ ) ( S dim τ+1 ). σ m 1 () τ m 2 () Through this identification, ι,l is a wedge of the restriction of ι rest (v),rest L (v) and Σι link (v),link L (v), where L is a shifted subcomplex of with v L. Proof. Let w be the second greatest vertex of. Notice that m 1 () ={σ m() w σ} and m 2 () ={τ w τ and τ w m()}. Then the first assertion follows. The second assertion is clear. 4. The space W n The purpose of this section is to introduce the space W n for a certain shifted complex and to show that it is built up inductively by two kinds of pushouts. Hereafter, we fix a collection of spaces X = {X i } n i=1. We also fix, L, M to be a shifted complex on the index set [n,n], a shifted subcomplex of on [n L,n] and a shifted subcomplex of L on [n M,n], respectively Definition of W n. For a shifted complex P on the index set V (P ) [m], we define WP m =( ΣP I X I ). I V (P ) j [m] V (P ) Then if n = 1, W n has the homotopy type of the right hand side of the homotopy equivalence in Theorem 1.4 since A B ΣA B and I I. We also define W m = m j=1. We next define a map,l : WL n Wn. Let δ i be the composite of maps ΣA X i (ΣA X i ) (ΣA X i ) ΣA (ΣA X i ), where is the suspension comultiplication and the second arrow is a wedge of the projections. Then δ i is a homotopy equivalence. We now define,l as follows. We first apply the projection WL n (( I [n L,n] ΣL I X I ) n L 1 j=n ) j<n and then δ nl 1, δ nl 2,...,δ n in turn to get a map WL n ( ΣL I X I J ). j<n I [n L,n] J [n,n L 1] We next apply the product of a wedge of Σι I J,L I 1 : ΣL I X I J Σ I J and the identity map of j<n to the target of the above map. Then we obtain a map,l : WL n Wn. We also define, : W n Wn as the composite proj W. n By definition, we have W n j<n,l L,M =,M, where M may be. We will often abbreviate,l by when, L are clear in the context. X I J

8 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS Pushouts in W n. Let us consider two kinds of pushouts by which Wn inductively. Define W n,c as a pushout link (n) i 1 C link (n) is constructed and a map c,l : Wn,c L Wn,c rest L (n) 1 rest (n) W n,c as the induced map from a commutative diagram 1 link L (n) i 1 C link L (n) 1 rest (n) 1 1 link (n) 1 i 1 C link (n). Proposition 4.1. Suppose is connected. Then there is a homotopy equivalence h c : Wn,c W n satisfying if L is also connected. Proof. Put W (i) = h c c,l =,L h c L I [n,n 1] ( σ m i ( I n ) ΣS dim σ X I ) for i = 0, 1, 2. Since rest I n (n) = rest (n) I and link I n (n) = link (n) I for I [n,n 1], it follows from Proposition 3.5 that W n 1 rest (n) =(W (0) W (1)) j<n, W n 1 link (n) =(W (0) W (2)) j<n and rest (n),link (n) is the product of the identity map of j<n and a wedge of the identity map of W (0) and the constant map. Put W c =(W (0) (W (1) X n ) (ΣW (2) X n )) j<n and r c,l : W c L W c to be the induced map from rest (n),rest L (n) and link (n),link (n). Then it follows from Proposition 3.5 that we can apply Lemma 2.4 to W n,c equivalence satisfying g c c,l = rc,l gc L. g c : W n,c W c and obtain a homotopy For I [n,n 1], rest I n (n) = I. Then by Proposition 3.5, W (0) W (1) = I [n,n 1] Σ I X I. On the other hand, (W (1) X n )(ΣW (2) X n )= =I [n,n 1] Σ I n X I n by Proposition 3.6. Then it follows that W n =(W (0) W (1) (W (1) X n ) (ΣW (2) X n )) j<n. Moreover, through this identification, one sees that,l is induced from rest (n),rest L (n) and link (n),link L (n) by Proposition 3.5 and 3.6. Since is connected, so is I n also by Proposition

9 8 OUYEMON IRIYE AND DAISUE ISHIMOTO 3.3. Then by Proposition 3.4, dim σ > 0 for σ m 1 ( I n ) and hence we can define a map δ n : X n W (1) W (1) (X n W (1)) by using the suspension parameter of S dim σ for σ m 1 ( I n ). We now define a homotopy equivalence h c W n,c g c W c =(W (0) (W (1) X n ) (ΣW (2) X n )) as the composite j<n (1δ n1) 1 (W (0) W (1) (W (1) X n ) (ΣW (2) X n )) = W n. j<n Since the suspension parameters used for r c,l and the above δ n are distinct, it holds that ((1 δ n 1) 1) r c,l =,L ((1 δ n 1) 1). Then one has h c c,l = ((1 δ n 1) 1) g c c,l = ((1 δ n 1) 1) r c,l g c L =,L ((1 δ n 1) 1) g c L =,L h c L, completing the proof. Define W n,d as a pushout W n W n CX n 1 j n 1 W n,d. When n L = n, we also define a map d,l : Wn,d L commutative diagram W n,d as the induced map from a W n L W n W n CX nl 1 j n L 1 W n CX n 1 j n 1. Proposition 4.2. There is a homotopy equivalence h d : Wn,d n L = n, Proof. Let δ i be the composite Σ(A X i ) (1 ) h d d,l = (n 1),L (n L 1) h d L. W n (n 1) 3 Σ(A Xi ) ΣA ΣX i (ΣA X i ), such that when

10 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 9 where the second arrow is a wedge of the projections. Then δ i is a homotopy equivalence. By applying δ m, δ m 1,..., δ i+2 in turn to ΣX i m j=i+1, we obtain a homotopy equivalence m δ m (i) :ΣX i Σ X I i = Σ{I 0,i} X I i, j=i+1 where I 0 is the maximum of I. Put W d = (( I [n,n] I [i+1,m] I [i+1,m] Σ I X I ) X n 1 (ΣX n 1 j n )) j<n 1 and r d,l : W d L W d to be the induced map from,l, where n L = n. Then by Lemma 2.4 and Proposition 3.5, there is a homotopy equivalence g d : W n,d W d satisfying g d d,l = rd,l gd L. Define hd as the composite of gd and a homotopy equivalence W d = (( Σ I X I ) X n 1 (ΣX n 1 )) =I [n,n] j n j<n 1 (δ n 1δ n (n 1)) 1 ( (Σ I X I ) (Σ( I {I 0,n 1}) X I (n 1) )) =( I [n,n] = W n (n 1), I [n,n] (Σ I X I ) (Σ I (n 1) X I (n 1) ) j<n 1 j<n 1 where the base point of {I 0,n 1} is I 0. Then since g d d,l = rd,l gd L and ((δ n 1 δ n (n 1)) 1) r d,l = (n 1),L (n L 1) ((δ nl 1 δ n (n L 1)) 1), one can see quite similarly to h c that Thus the proof is completed. h d d,l = (n 1),L (n L 1) h d L. Let us show further properties of the homotopy equivalence h d. Proposition 4.3. Define a map µ,l : WL n Wn,d diagram W n L W n as the induced map from the commutative W n W n W n CX n 1 j n 1. Then h d µ,l = (n 1),L. Proof. By the definition of h d, the statement is true for µ,. Since µ,,l = µ,l, the proof is completed.

11 10 OUYEMON IRIYE AND DAISUE ISHIMOTO We put S m (i, k) =( =I [i+1,m] J [k,i 1] Σ{I 0,i} X I J i ) j<k. Let us define a map δ m (i, k) :CX i j [m] i S m (i, k) for 1 k i<m. We first apply the projection CX i j [m] i (ΣX i m j=i+1 ) j<i and then a homotopy equivalence δ m (i) to get a map CX i S m (i, i). j [m] i We next apply the projection S m (i, i) (( =I [i+1,m] Σ{I 0,i} X I i ) i 1 j=k ) j<k and then δ i 1, δ i 2,...,δ k in turn to obtain δ m (i, k). Let ι (i) :S n (i, n ) W n be the product of the identity map of j<n and a wedge of Σι I J i,{i 0,i} 1:Σ{I 0,i} X I J i Σ I J i X I J i for = I [i +1,n],J [n,i 1]. We now define for n i<nand θ (n) as the composite CX i j i θ (i) =ι (i) δ n (i, n ) proj j<n W n. Let us make a list of properties of the map θ (i). Let i : CX i CX i be the composite CX i CX i ΣX i where the first arrow is the co-action map. proj CX i, Proposition 4.4. The map θ (i) has the following properties. (1) θ (i) restricts to, for n i n. (2) θ (i) =,L θ L (i) for n L i n. (3) If is connected, the composite X n CX i j i,n 1 i 1 X n CX i j i,n 1 θ rest (n)(i) rest (n) W n is equal to θ (i) for n i < n, where is the composite of the canonical map rest (n) Wn,c and hc. (4) The composite CX n 1 j n 1 W n,d h d W n (n 1) is equal to θ (n 1)(n 1), where the first arrow is the canonical map.

12 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 11 Proof. 1, 2 and 4 follow immediately from the definitions of θ (i) and the homotopy equivalence h d. We prove 3. By the definition of, it is true for θ (n) if n = n 1. Let n i<n 1. Notice that by Proposition 3.3, ι (i) maps S n (i, n ) into ( I Σ I X I ) j<n W n, where I ranges over all nonempty subsets of [n,n] such that I is discrete. It follows from Proposition 3.4 that there is a map S n 1 (i, n ) W (0) j<n k satisfying a commutative diagram S n (i, n ) W n ι (i) proj S n 1 (i, n ) S n 1 (i, n ) W (0) j<n k W n 1 ι rest (n)(i) rest (n). Then since the restriction of to X n (W (0) j<n ) X n W (0) j<n k we get a commutative diagram proj W (0) j<n k W n rest (n) is the composite as in the proof of Proposition 4.1, X n S n 1 (i, n ) 1 ι rest (n)(i) rest (n) proj S n 1 (i, n ) S n (i, n ) ι (i) W n. On the other hand, there is a commutative diagram X n CX i j i,n 1 i 1 X n CX i j i,n δ n (i,n ) 1 δ n 1 (i,n ) S n (i, n ) proj X n S n 1 (i, n ) proj S n 1 (i, n ). Then since ι (i) is equal to the composite S n (i, n ) proj S n 1 (i, n ) S n (i, n ) ι (i) W n as above, we obtain the desired result. Proposition 4.5. Suppose n <n L and define a map ν,l : W n,d L from a commutative diagram Wn as the induced map W n L W n CX nl 1 j n L 1 W n W n W n. θ (n L 1) Then ν,l =,L (nl 1) h d L. Proof. By the definition of h d L, ν L,L = h d L. Since θ (n L 1) =,L (nl 1) θ L (n L 1) by Proposition 4.4, it holds that ν,l =,L (nl 1) ν L,L, completing the proof.

13 12 OUYEMON IRIYE AND DAISUE ISHIMOTO 5. Decomposition of Z n In this section, we introduce the space Z n corollary, we obtain Theorem 1.4. and prove its decomposition. As an immediate 5.1. The space Z n. For a simplicial complex P on the index set V (P ) [m], we define ZP m = ( C ). σ P j σ j [m] σ Then if n = 1, Z n = Z (CX,X). When Q is a subcomplex of P, we denote the usion Z m Q Zm P by P,Q. We also put Z m = m j=1 and P, to be the usion Z m Zm P. Notice that P,Q is a cofibration, where Q may be. Likewise,L, we will often abbreviate,l by when, L are clear. Proposition 5.1. There are pushouts X n Z n 1 link (n) i 1 CX n Z n 1 link (n) and Z n CX n 1 j n 1 1 X n Z n 1 rest (n) Z n Z n Z n (n 1), where i : X n CX n is the usion. Proof. There is a pushout of simplicial complexes link (n) rest (n) star (n) resulting a pushout of topological spaces Z n link (n) Z n rest (n) Z n star (n) Z n. We have ZP n = X n Z n 1 P for P = link (n), rest (n) and Zstar n (n) = CX n Z n 1 link (n). Through this identification, we obtain the first pushout. The second pushout is obtained quite similarly Main theorem. We now state the main result. Theorem 5.2. There is a homotopy equivalence ɛ : Z n where L may be. W n such that ɛ,l =,L ɛ L,

14 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 13 Proof. First of all, we put ɛ : Z n Wn to be the identity map of n j=1. By induction on n, we construct ɛ satisfying ɛ,l =,L ɛ L and ɛ,i = θ (i). We will abbreviate ɛ by ɛ when is clear in the context. For n = 1, must be 1. We put ɛ : Z 1 = CX 1 = W 1 to be the constant map and then it follows that ɛ, =, ɛ and ɛ,1 = θ (1). Assume that the theorem holds less than n. Let 0,L 0 be the connected components of, L containing the vertex n, respectively. Then by Proposition 3.3, V ( 0 )=[n 0,n] and V (L 0 )=[n L0,n] for some n 0 [n,n] and n L0 [n L,n]. Put (m) = 0, [m,n], according as n 0 m n, n m<n 0, 1 m<n. We construct ɛ (m) by induction on m. By the hypothesis of induction on n, there is a commutative diagram X n Z n 1 rest 0 (n) 1 X n Z n 1 link 0 (n) i 1 CX n Z n 1 link 0 (n) 1 ɛ rest 0 (n) 1 1 ɛ link 0 (n) 1 ɛ i 1 C link 0 (n). By Proposition 5.1, the pushout of the top row is Z n 0. Then we define a homotopy equivalence ɛ 0 : Z n 0 W n,c 0 as the induced map from the above diagram. By the induction hypothesis, there is also a commutative diagram X n Z n 1 rest L0 (n) 1 ɛ 1 1 X n Z n 1 i 1 CX n Z n 1 link L0 (n) 1 ɛ link L0 (n) 1 ɛ rest L0 (n) 1 link L0 (n) i 1 C link L0 (n) X n Z n 1 1 rest 0 (n) 1 ɛ 1 rest 0 (n) 1 X n Z n 1 link 0 (n) 1 ɛ i CX n Z n 1 link 0 (n) 1 ɛ 1 1 i 1 C link 0 (n) link 0 (n). Then it holds that ɛ 0 0,L 0 = c 0,L 0 ɛ L0. Let : Z n Zn be the restriction of n j=1 j : n j=1 C n j=1 C. Then,L =,L L. By Proposition 4.1, if we put ɛ 0 = h c 0 ɛ 0 0, it holds that ɛ 0 0,L 0 = 0,L 0 ɛ L0,

15 14 OUYEMON IRIYE AND DAISUE ISHIMOTO where L may be. Moreover, by the induction hypothesis and Proposition 4.4, we have ɛ 0 0,i = h c 0 ɛ 0 0 0,i = h c 0 ɛ 0 0,i i = (1 (ɛ rest0 (n) rest0 (n),i)) i = (1 θ rest0 (n)(i)) i = θ 0 (i) for n 0 i<n, where is the composite of the canonical map rest 0 (n) Wn,c 0 as above. We also have h c 0 ɛ 0 0,n = (1 link0 (n), ) n, where is the composite of the canonical map C link 0 (n) Wn,c 0 and h c 0. By the definition of h c 0, the right hand side is the composite CX n j<n X proj j j<n 0 W n 0, implying ɛ 0 0, = θ 0 (n). Summarizing, we have constructed a homotopy equivalence ɛ 0 = ɛ (n) : Z(n) n W(n) n satisfying for n 0 i n. ɛ (n) (n),l(n) = (n),l(n) ɛ L(n) and ɛ (n) (n),i = θ (n) (i) Put (Z(m+1) n, Zn (m+1), Wn (m+1), Wn (m+1) ) n 0 m<n (P (m),q (m), P (m), Q (m)) = (Z n,cx m j m, W n,cx m j m ) n m<n 0 (Z n, Zn, Wn, Wn ) 1 m<n. We also put P (m) = Q (m) = P (m) = Q (m) = n j=1. Then by the hypothesis of induction on m, there is a commutative diagram and Z n (m+1) P (m) Q (m) ɛ W n (m+1) ɛ P (m) ˆɛ Q (m), where ˆɛ : Q (m) Q (m) is defined as ɛ (m+1), the identity map of CX n j n and ɛ according as n 0 m < n, n m<n 0 and 1 m<n. We define a homotopy equivalence ɛ (m) as the induced map from the above commutative diagram for n 0 m<nand 1 m< n. Then ɛ (m) = ɛ 0, ɛ according as n 0 m < n, 1 m<n. By Proposition 4.2 and 5.1, we also define ɛ (m) as the composite of the induced map Z(m) n W n,d (m) from the above commutative diagram and h d (m) for n m<n 0. Then by the induction hypothesis and

16 DECOMPOSITIONS OF POLYHEDRAL PRODUCTS 15 Proposition 4.4, ɛ (m) (m),i = ɛ (m) (m),(m+1) (m+1),i = (m),(m+1) ɛ (m+1) (m+1),i = (m),(m+1) θ (m+1) (i) = θ (m) (i) for min{n 0,m+1} i n. By Proposition 4.4, one also has ɛ (m) (m),m = θ (m) (m) if m<n 0. Define φ : Q L (m) Q (m) as (m+1),l(m+1), θ (m+1) (m) and the usion according as (Q L (m),q (m)) = (ZL(m+1) n, Zn (m+1) ), (CX m j m, Z(m+1) n ) and otherwise. We also define φ : Q L (m) Q (m) similarly using (m+1),l(m+1) and θ (m+1) (m). By the hypothesis of induction on m, there is a commutative diagram Z n L(m+1) ɛ P L (m) ɛ Q L (m) ˆɛ W n L(m+1) P L (m) φ Q L (m) Z n (m+1) ɛ W(m+1) n P (m) ɛ P (m) Q (m) φ ˆɛ Q (m). Then it follows from Proposition 4.2, 4.3 and 4.5 that ɛ (m) (m),l(m) = (m),l(m) ɛ L(m). Therefore we complete the proof. Proof of Theorem 1.4. As is noted above, if n = 1, Z n = Z (CX,X) and W n I [n] I X I. Then the result follows from Theorem 5.2. References [B] I. Baskakov, Cohomology of -powers of spaces and the combinatorics of simplicial divisions, Russian Math. Surveys 57 (2002), no. 5, [BBCG] A. Bahri, M. Bendersky, F. R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Advances in Math. 225 (2010), [BP] V.M. Buchstaber and T.E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series 24, American Mathematical Society, Providence, RI, [DJ] M.W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Mathematical Journal 62 (1991), [DO] M.W. Davis and B. Okun, Cohomology computations for Artin groups, Bestvina-Brady groups, and graph products, preprint, 2010.

17 16 OUYEMON IRIYE AND DAISUE ISHIMOTO [DS] G. Denham and A. Suciu, Moment-angle complexes, monomial ideals and Massey products, Pure Appl. Math. Q. 3 (2007), no. 1, [FT] Y. Félix and D. Tanré, Rational homotopy of the polyhedral product functor, Proc. Amer. Math. Soc. 137 (2009), no. 3, [GT] J. Grbić and S. Theriault, The homotopy type of the complement of a coordinate subspace arrangement, Topology 46 (2007), [H] P.S. Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, Providence, RI, [L] J. Lillig, A union theorem for cofibrations, Arch. Math. 24 (1973), [N] D. Notbohm, Colorings of simplicial complexes and vector bundles over Davis-Januszkiewicz spaces, Math. Z. 266 (2010), no. 2, [P] T. Porter, Higher-order Whitehead products, Topology 3 (1965), Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, , Japan address: kiriye@mi.s.osakafu-u.ac.jp Department of Mathematics, yoto University, yoto, , Japan address: kishi@math.kyoto-u.ac.jp