The homotopy types of moment-angle complexes for flag complexes

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 ZK 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 ZK 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 ZK 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 Ω ZK and ΩD-1muJ(K) are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime p≠ 2.