Polyhedral products for simplicial complexes with minimal Taylor resolutions

TL;DR

This paper establishes equivalences among algebraic and topological properties of simplicial complexes with minimal Taylor resolutions, linking Golodness, homotopy types, and polyhedral product decompositions.

Contribution

It proves that for such complexes, Golodness, the strong gcd-condition, and certain homotopy decompositions are equivalent, unifying algebraic and topological perspectives.

Findings

$K$ satisfies the strong gcd-condition if and only if it is Golod.

The moment-angle complex $\\mathcal{Z}_K$ is homotopy equivalent to a wedge of spheres.

The desuspension of the polyhedral product decomposition holds for these complexes.

Abstract

We prove that for a simplicial complex $K$ whose Taylor resolution for the Stanley-Reisner ring is minimal, the following four conditions are equivalent: (1) $K$ satisfies the strong gcd-condition; (2) $K$ is Golod; (3) the moment-angle complex $\mathcal{Z}_K$ is homotopy equivalent to a wedge of spheres; (4) the decomposition of the suspension of the polyhedral product $\mathcal{Z}_K(C\underline{X},\underline{X})$ due to Bahri, Bendersky, Cohen, and Gitler desuspends.