Fat wedge filtrations and decomposition of polyhedral products

TL;DR

This paper investigates the fat wedge filtration of polyhedral products, providing conditions for their decomposition, and explores connections with Golodness and properties of moment-angle complexes.

Contribution

It introduces a sufficient condition for decomposing polyhedral products via fat wedge filtration and links this to Golodness and Cohen-Macaulay properties.

Findings

Decomposition condition for polyhedral products based on fat wedge filtration.

Connection between decomposition and Golodness of the simplicial complex.

Necessary and sufficient conditions for moment-angle complexes to decompose and have co-H-structures.

Abstract

The polyhedral product constructed from a collection of pairs of cones and their bases and a simplicial complex $K$ is studied by investigating its filtration called the fat wedge filtration. We give a sufficient condition for decomposing the polyhedral product in terms of the fat wedge filtration of the real moment-angle complex for $K$, which is a desuspension of the decomposition of the suspension of the polyhedral product due to Bahri, Bendersky, Cohen, and Gitler. We show that the condition also implies a strong connection with the Golodness of $K$, and is satisfied when $K$ is dual sequentially Cohen-Macaulay over $\mathbb{Z}$ or $\lceil\frac{\dim K}{2}\rceil$-neighborly so that the polyhedral product decomposes. Specializing to moment-angle complexes, we also give a necessary and sufficient condition for their decomposition and co-H-structures in terms of their fat wedge filtration.