Homotopy types of moment-angle complexes associated to (almost) linear resolutions
The moment-angle complex associated to a simplicial complex K is a central object of study in toric topology which controls the homotopy groups of all toric manifolds. In this talk, we consider the problem of reading off the homotopy type of these spaces from homological properties of their associated Stanley-Reisner rings. We show that the Hurewicz image for any moment-angle complex contains the linear strand of its associated Stanley-Reisner ring. Combined with variants of a theorem of Eagon and Reiner well-known to commutative algebraists, we describe how this recovers results of various authors identifying the homotopy type of the moment-angle complex as a wedge of spheres when the Stanley-Reisner ring satisfies certain linearity properties. Going further, we introduce a large class of Gorenstein simplicial complexes whose associated moment-angle manifolds are formal, having the rational homotopy type of connected sums of sphere products and the (integral) loop space homotopy type of products of loops on spheres. This is joint work with Ben Briggs.