The polyhedral product functor: a method of computation for moment ...

arXiv:0711.4689v2 [math.AT] 8 Dec 2008

THE POLYHEDRAL PRODUCT FUNCTOR: A METHOD OF COMPUTATION FOR MOMENT-ANGLE COMPLEXES, ARRANGEMENTS AND RELATED SPACES A. BAHRI, M. BENDERSKY, F. R. COHEN, AND S. GITLER Abstract. This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson [20], Hochster[22], Baskakov [3], Panov [36], and Buchstaber-Panov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of Denham-Suciu in [16].

1. Introduction Spaces which now are called (generalized) moment-angle complexes, have been studied by topologists since the 1960’s thesis of G. Porter [39]. In the 1970’s E. B. Vinberg [45] and in the late 1980’s S. Lopez de Medrano developed some of their features [30]. In seminal work during the early 1990’s, M. Davis and J. Januszkiewicz introduced quasi-toric manifolds, a topological generalization of projective toric varieties which were being studied intensively by algebraic geometers [12]. They observed that every quasi-toric manifold is the quotient of a moment-angle complex by the free action of a real torus. The moment-angle complex is denoted Z(K; (D 2, S 1 )). The integral cohomology of the spaces Z(K; (D 2 , S 1 )) has been studied by GoreskyMacPherson [20], Hochster [22], Baskakov [3], Panov [36], Buchstaber-Panov [7] and Franz [17, 18]. Among others who have worked extensively on generalized moment-angle complexes 1991 Mathematics Subject Classification. Primary: 13F55, 14F45, 32S22, 52C35, 55U10 Secondary: 16E05, 57R19. Key words and phrases. arrangements, cohomology ring, moment-angle complex, simplicial complex, simplicial sets, stable splittings, Stanley-Reisner ring, suspensions, toric varieties. 1

THE POLYHEDRAL PRODUCT FUNCTOR

2

are Notbohm-Ray [35], Grbic-Theriault [21], Strickland [43] and Kamiyama-Tsukuda [27]. The direction of this paper is guided by the development in elegant work of Denham-Suciu [16]. Among the results given here is a natural decomposition for the suspension of the generalized moment-angle complex, the value of the suspension of the “polyhedral product functor”. Since the decomposition is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This last decomposition specializes to the homological decompositions in the work of several authors cited above. Furthermore, this decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex extended to other natural settings and which arises from generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. Extensions of these structures to a simplicial setting as well as to the structure of ordered, commuting n-tuples in a Lie group G are given in [1]. Generalized moment-angle complexes are given by functors determined by simplicial complexes with values given by subspaces of products of topological spaces. Thus, these may be regarded as “polyhedral product functors” indexed by abstract simplicial complexes, terminology for which the authors thank Bill Browder (also the inventor of the name “orbifold”).

Contents 1. Introduction 2. Statement of results 3. Toric manifolds 4. Preparation 5. Proof of Theorem 2.10 6. Proof of Theorem 2.12 7. Proof of Theorem 2.19 8. Proof of Theorem 2.21 9. Proof of Theorem 2.34 10. Proof of Theorem 2.45 11. Connection to Kurosh’s theorem 12. Acknowledgements References

1 3 19 23 26 28 30 31 32 33 34 36 36

THE POLYHEDRAL PRODUCT FUNCTOR

3

2. Statement of results In this article, a polyhedron is defined to be the geometric realization of a simplicial complex. Generalized moment-angle complexes can be be regarded loosely as a functor from simplicial complexes with values given by subspaces of products of topological spaces. Thus these may be regarded as ”polyhedral product functors”. Generalized moment-angle complexes are defined next. (1) Let (X, A) = {(Xi , Ai , xi )m i=1 } denote a set of triples of CW –complexes with basepoint xi in Ai . (2) Let K denote an abstract simplicial complex with m vertices labeled by the set [m] = {1, 2, . . . , m}. Thus, a (k − 1)-simplex σ of K is given by an ordered sequence σ = (i1 , . . . , ik ) with 1 ≤ i1 < . . . < ik ≤ m such that if τ ⊂ σ, then τ is a simplex of K. In particular the empty set φ is a subset of σ and so it is in K. The set [m] is minimal in the sense that every i ∈ [m] belongs to at least one simplex. The length of of σ = k is denoted |σ|. (3) Let ∆[m − 1]q denote the q-skeleton of (m − 1)-simplex having m vertices. That is, ∆[m − 1]q denotes the subsets of [m] of cardinality at most q + 1. Define two functors from the category K of simplicial complexes with morphisms simplicial maps to the category CW∗ of connected, based CW -complexes and based continuous maps as follows. Definition 2.1. As above, let (X, A) denote the collection {(Xi , Ai , xi )}m i=1 . The generalized moment-angle complex or polyhedral product functor determined by (X, A) and K denoted Z(K; (X, A)) is defined using the functor D : K → CW∗ as follows: For every σ in K, let D(σ) =

m Y

Yi ,

where Yi =

i=1

(

Xi if i ∈ σ Ai if i ∈ [m] − σ.

with D(∅) = A1 × . . . × Am . The generalized moment-angle complex is [ D(σ) = colimD(σ) Z(K; (X, A)) = σ∈K

THE POLYHEDRAL PRODUCT FUNCTOR

4

where the colim is defined by the inclusions, dσ,τ with σ ⊂ τ and D(σ) is topologized as a subspace of the product X1 × . . . × Xm . The generalized pointed moment-angle is the underlying space Z(K; (X, A)) with base-point ∗ = (x1 , . . . , xm ) ∈ Z(K; (X, A)). Note that the definition of Z(K; (X, A)) did not require spaces to be either based or CW complexes. In the special case where Xi = X and Ai = A for all 1 ≤ i ≤ m, it is convenient to denote the generalized moment-angle complex by Z(K; (X, A)) to coincide with the notation in [16]. The smash product X1 ∧ X2 ∧ . . . ∧ Xm is given by the quotient space (X1 × . . . × Xm )/S(X1 × . . . × Xm ) where S(X1 × . . . × Xm ) is the subspace of the product with at least one coordinate given by the base-point xj ∈ Xj . Spaces analogous to generalized moment-angle complexes are given next where products of spaces are replaced by smash products, a setting in which base-points are required. Definition 2.2. Given a generalized pointed-moment-angle complex Z(K; (X, A)) obtained from (X, A, ∗), the generalized smash moment-angle complex b Z(K; (X, A))

is defined to be the image of Z(K; (X, A)) in the smash product X1 ∧ X2 ∧ . . . ∧ Xm . b b and is The image of D(σ) in Z(K; (X, A)) is denoted by D(σ) Y1 ∧ Y2 ∧ . . . ∧ Ym

where Yi =

(

Xi if i ∈ σ Ai if i ∈ [m] − σ.

b As in the case of Z(K; (X, A)), note that Z(K; (X, A)) is the colimit obtained from the b b b b spaces D(σ) with D(σ) ∩ D(τ ) = D(σ ∩ τ ).

b (X, A)) are bifunctors, natural for Remark 2.3. The constructions Z(K; (X, A)) and Z(K; morphisms of (X, A) as well as (X, A, ∗), and for injections in K. The next example provides an elementary case of a moment-angle complex together with an associated toric manifold. Example 2.4. Let K denote the simplicial complex given by {{1}, {2}, ∅} with (Xi , Ai ) = (D ni , S ni −1 ) for i = 1, 2, and let K ′ denote an arbitrary simplicial complex with m vertices.

THE POLYHEDRAL PRODUCT FUNCTOR

5

(1) The first example is (Xi , Ai ) = (D 2n , S 2n−1 ) for i = 1, 2 where D 2n is the 2n-disk and S 2n−1 is its boundary sphere. Then Z(K; (D 2n , S 2n−1 )) = D 2n × S 2n−1 ∪ S 2n−1 × D 2n with the boundary of D 2n × D 2n given by ∂(D 2n × D 2n ) = D 2n × S 2n−1 ∪ S 2n−1 × D 2n so Z(K; (D 2n , S 2n−1 )) = S 4n−1 . (2) The second example is (X1 , A1 ) = (D 2 , S 1 ) and (X2 , A2 ) = (D 3 , S 2 ). Then Z(K; (X, A)) = D 2 × S 2 ∪ S 1 × D 3 = S 4 . (3) The third example exhibits the connection to toric manifolds where K ′ is a simplicial complex with m vertices. Regard (D 2n , S 2n−1 ) as a pair of subspaces of Cn in the standard way with the standard S 1 -action restricting to an action on the pair (D 2n , S 2n−1 ). Thus T m acts on Z(K ′ ; (D 2n , S 2n−1 )). Specialize to K ′ = K. Notice that the S 1 diagonal subgroup ∆(T 2 ) acts freely on Z(K; (D 2n , S 2n−1 )) = S 4n−1 where the quotient space S 4n−1 /∆(T m ) is CP2n−1 , complex projective space of dimension (2n − 1). If n = 1, then (D 2n , S 2n−1 ) = (D 2 , S 1 ) and Z(K; (D 2 , S 1 )) = S 3 is a case related to toric topology. That is, CP1 is the toric manifold quotient by ∆(T 2 ) of its associated moment-angle complex. (4) More generally, let ∆[m − 1]q be the q-skeleton of the (m − 1)-simplex ∆[m − 1] consisting of all the simplices of ∆[m − 1] of dimension ≤ q. Since ∂∆[m − 1] = ∆[m − 1]m−2 , there is a homeomorphism Z(∂∆[m − 1]; (D 2n , S 2n−1 )) → S 2mn−1 and again the diagonal group ∆(T m ) = S 1 acts freely on S 2mn−1 with quotient the manifold CPmn−1 . Again when n = 1, there is a homeomorphism Z(∂∆[m − 1]; (D 2, S 1 )) → S 2m−1 and the quotient by the action of the diagonal subgroup is CPm−1 , a description given by CPm−1 as a toric manifold. A stronger relation between the complex K ′ and a toric manifold M 2n is given in Section 3.

THE POLYHEDRAL PRODUCT FUNCTOR

6

Definition 2.5. Consider the ordered sequence I = (i1 , . . . , ik ) with 1 ≤ i1 < . . . < ik ≤ m together with pointed spaces Y1 , . . . , Ym . Then (1) (2) (3) (4) (5)

the length of I is |I| = k, the notation I ≤ [m] means I is any subsequence of (1, . . . , m), Y [m] = Y1 × . . . × Ym , Y I = Yi1 × Yi2 × . . . × Yik , bI denotes Ai1 ∧ . . . ∧ Ai for I = (i1 , . . . , ik ). Yb I = Yi1 ∧ . . . ∧ Yik , and (redundantly) A k

Two more conventions are listed next.

(1) The symbol X ∗ Y denotes the join of two topological spaces X and Y . If X and Y are pointed spaces of the homotopy type of a CW-complex, then X ∗ Y has the homotopy type of the suspension Σ(X ∧ Y ). (2) Given the family of pairs (X, A) = {(Xi , Ai )}m i=1 and I ≤ [m], define j=|I|

(XI , AI ) = {(Xij , Aij )}j=1 which is the subfamily of (X, A) determined by I.

Standard constructions for simplicial complexes and associated posets are recalled next. Definition 2.6. Let K denote a simplicial complex with m vertices. (1) Recall that the empty simplex φ is required to be in K. (2) Given a sequence I = (i1 , . . . , ik ) with 1 ≤ i1 < . . . < ik ≤ m, define KI ⊆ K to be the full sub-complex of K consisting of all simplices of K which have all of their vertices in I, that is KI = {σ ∩ I|σ ∈ K}. (3) Let |K| denote the geometric realization of the simplicial complex K. ¯ (4) Associated to a simplicial complex K, there is a partially ordered set (poset) K ¯ corresponds to a simplex σ ∈ K with order given given as follows. A point σ in K ¯ if and only if σ2 ⊆ σ1 in K. by reverse inclusion of simplices. Thus σ1 ≤ σ2 in K ¯ The empty simplex φ is the unique maximal element of K. Let P be a poset with p ∈ P . There are further posets given by P≤p = {q ∈ P |q ≤ p} as well as P