Homotopy exponents of polyhedral products

TL;DR

This paper investigates homotopy exponents and Moore's conjecture for polyhedral products, establishing conditions under which these spaces lack homotopy exponents at various primes and confirming Moore's conjecture in specific cases.

Contribution

It proves that rationally hyperbolic polyhedral products with torsion-free homology have no homotopy exponent at odd primes and verifies Moore's conjecture under certain conditions.

Findings

Polyhedral products with torsion-free homology and rational hyperbolicity lack homotopy exponents at odd primes.

Moore's conjecture holds for polyhedral products where suspensions are wedges of simply-connected spheres.

Criteria are provided for polyhedral join products to be hyperbolic and to lack homotopy exponents at many primes.

Abstract

We study Moore's conjecture and homotopy exponents for polyhedral products. For $(\underline{CA},\underline{A})^K$ where each $A_i$ is finite and has torsion-free homology, we prove that if $(\underline{CA},\underline{A})^K$ is rationally hyperbolic, then it has no homotopy exponent at any odd prime. Under the additional hypothesis $\Sigma A_i$ is homotopy equivalent to a finite-type wedge of simply-connected spheres, we show Moore's conjecture holds for $(\underline{CA},\underline{A})^K$. We also give criteria such that, for a large family of polyhedral join products, the associated polyhedral products are rationally hyperbolic, mod-$p^r$ hyperbolic for all but finitely many primes, and have no homotopy exponent at all but finitely many primes.