Anick's conjecture for polyhedral products

Abstract

We develop a method for studying the pointed loop space of general polyhedral products, showing that many properties are determined by the moment-angle complex. To apply the method, we show that localised away from a finite set of primes, the loop space of a moment-angle complex is homotopy equivalent to a product of loops on spheres. As a consequence, we give p-local loop space decompositions of quasitoric manifolds, certain toric orbifolds and a wide family of polyhedral products. This verifies a conjecture of Anick for such spaces. We also describe the additive structure of loop homology of simply connected polyhedral products in terms of polynomials studied by Backelin and Berglund.

26 pages, improved torsion and Steenrod algebra results