Lewis Stanton: Anick's conjecture for polyhedral products

Abstract: Anick conjectured the following after localisation at any
sufficiently large prime - the pointed loop space of any finite, simply
connected CW complex is homotopy equivalent to a finite type product of
spheres, loops on spheres, and a list of well-studied torsion spaces
defined by Cohen, Moore and Neisendorfer. We study this question in the
context of moment-angle complexes, a central object in toric topology
which are indexed by simplicial complexes. These are a special case of a
family of spaces known as polyhedral products, which unify constructions
across mathematics. Recently, much work has been done to find families
of simplicial complexes for which the corresponding moment-angle complex
satisfies Anick's conjecture integrally. In this talk, I will survey
what is known, and show that the loop space of any moment-angle complex
is homotopy equivalent to a product of looped spheres after localisation
away from a finite set of primes. This is then used to show Anick's
conjecture holds for a much wider family of polyhedral products. This
talk is based on joint work with Fedor Vylegzhanin.