Cellular Cochain Algebras and Torus Actions

We prove that the integral cohomology algebra of the moment-angle complex ZK [1] is isomorphic to the Tor-algebra of the face ring of simplicial complex K. The proof relies upon the construction of a cellular approximation of the diagonal map ∆: ZK → ZK × ZK . Cellular cochains do not admit a functorial associative multiplication because a proper cellular diagonal approximation does not exist in general. The construction of momentangle complexes is a functor from the category of simplicial complexes to the category of spaces with torus action. We show that in this special case the proposed cellular approximation of the diagonal is associative and functorial with respect to those maps of moment-angle complexes which are induced by simplicial maps. The face ring of a complex K on the vertex set [m] = {1, . . . ,m} is the graded quotient ring Z[K] = Z[v1, . . . , vm]/(vω : ω / ∈ K) with deg vi = 2 and vω = vi1 · · · vik , where ω = {i1, . . . , ik} ⊆ [m]. Let BT m be the classifying space for the m-dimensional torus, endowed with the standard cell decomposition. Consider a cellular subcomplex DJ (K) := ⋃ σ∈K BT σ ⊆ BT, where BT σ = {x = (x1, . . . , xm) ∈ BT m : xi = pt for i / ∈ σ}. Using the cellular decomposition we establish a ring isomorphism H(DJ (K)) ∼= Z[K] (see [2, Lemma 2.8]). Let D2 ⊂ C be the unit disc and set Bω := {(z1, . . . , zm) ∈ (D 2)m : |zj | = 1 for j / ∈ ω}. The moment-angle complex is the T-invariant subspace ZK := ⋃ σ∈K Bσ ⊆ (D 2)m. As it is shown in [1, Ch. 6], the spaces DJ (K) and ZK are homotopy equivalent to the spaces introduced in [3], which justifies our notation. Complexes ZK provide an important class of torus actions. The space ZK is the homotopy fibre of the inclusion DJ (K) →֒ BT; it appears also as the level surface of the moment map used in the construction of toric varieties via symplectic reduction; it is also homotopy equivalent to the complement of the coordinate subspace arrangement determined by K, see [1, § 8.2].