Diagram Categories in Homotopy Theory CRG Seminar: Steven Amelotte

Toric topology assigns to each simplicial complex K a space with a torus action, called the moment-angle complex, which is defined as a polyhedral product or (homotopy) colimit over the face category of K. These spaces play a universal role in toric topology and control the homotopy groups of all toric manifolds. In this talk, we consider the problem of reading off the homotopy types of these spaces from homological properties of their associated Stanley-Reisner rings. In particular, we show that the Hurewicz image of any moment-angle complex contains the linear strand of the corresponding Stanley-Reisner ideal, and describe how this can be combined with some well-known results in commutative algebra to analyze the formality and homotopy type of a large class of moment-angle manifolds and their loop spaces. This talk is based on joint work with Ben Briggs.