The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures.A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.2000 Mathematics Subject Classification. Primary 13F55, 55P15, Secondary 52C35.

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jöllenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop spaces ΩZ K and ΩDJ(K) are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime p = 2.

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X 1 , . . . , X n are pointed connected C W -complexes and C X i is the cone on X i , then the polyhedral product determined by K and the pairs (C X i , X i ) is homotopy equivalent to a wedge of suspensions of smashes of the X i 's. Earlier work of the authors dealt with the special case where each X i is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.

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This paper aims to find the most general combinatorial conditions under which a moment-angle complex (D 2 , S 1 ) K is a co-H-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment-angle complex over a Golod simplicial complex is a co-H-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces.