Anick's conjecture for polyhedral products
Lewis Stanton, Fedor Vylegzhanin
TL;DR
The paper develops a comprehensive $p$-local framework to analyze ΩX for polyhedral products by reducing to moment-angle complexes, proving that localizing away from finitely many primes yields a product decomposition into spheres and indecomposable torsion spaces. Central to the approach are Anick's $R$-local Milnor–Moore and Hilton–Serre–Baues theorems, together with the Backelin–Berglund polynomial machinery that encodes loop-homology via Stanley–Reisner rings. The authors certify Anick's conjecture for a broad family of spaces arising from polyhedral products, including moment-angle complexes, partial quotients, and simply connected toric orbifolds, and provide explicit p-local decompositions and Poincaré series formulas. These results connect combinatorial data of simplicial complexes to the homotopy types of toric-related spaces, enabling practical computations of loop spaces and their homotopy groups in the localized setting. Overall, the work extends Anick’s conjecture to a wide class of polyhedral products and clarifies the role of moment-angle complexes as a universal source of $p$-local loop-space structure in toric topology.
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.