Fat-wedge filtration and decomposition of polyhedral products

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 it is satisfied when $K$ is dual sequentially Cohen–Macaulay over $\mathbb{Z}$ or $\lceil \frac{dimK}{2}\rceil$-neighborly so that the polyhedral product decomposes. Specializing to the moment-angle complex, we prove that the similar condition on its fat-wedge filtrations is necessary and sufficient for its decomposition.