Anick's conjecture for polyhedral products

TL;DR

This paper proves Anick's conjecture for a broad class of polyhedral products by analyzing their loop spaces, revealing homotopy equivalences and decompositions that advance understanding in algebraic topology.

Contribution

It introduces a new method for studying loop spaces of polyhedral products and verifies Anick's conjecture for various classes of these spaces.

Findings

Loop space of moment-angle complexes decomposes into products of spheres after localization.

p-local decompositions of quasitoric manifolds and toric orbifolds are established.

Additive structure of loop homology described using special polynomials.

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.