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.