Anick's conjecture for polyhedral products

2025/06/18 by Stanton, Lewis, Vylegzhanin, Fedor
#14M25 #16E30 #55P60 #57R18 #57R19 #57S12 #Algebraic Topology (math.AT) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 55P35 #Secondary 13F55

paper · doi:10.48550/arxiv.2506.15573

Abstract

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. This verifies a conjecture of Anick for such spaces. We also develop a method to reduce the study of the loops of more general polyhedral products to that of the loops on a moment-angle complex. As a consequence, we give p-local loop space decompositions of quasitoric manifolds, certain toric orbifolds and a wide family of polyhedral products. We also describe the additive structure of loop homology of simply connected polyhedral products in terms of polynomials studied by Backelin and Berglund.

Citations

• Loop spaces of n-dimensional Poincaré duality complexes whose (n-1)-skeleton is a co-H-space
• Cohomology bases of toric surfaces
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• Homotopy groups of highly connected manifolds
• The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations
• The threshold for integer homology in random d-complexes
• Loops on polyhedral products and diagonal arrangements
• Differential posets and Smith normal forms

Related

• Loop spaces of polyhedral products associated with the polyhedral join product
• Polyhedral products associated to pseudomanifolds
• Polyhedral products and features of their homotopy theory
• Loops on polyhedral products and diagonal arrangements
• Operations on polyhedral products and a new topological construction of infinite families of toric manifolds