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

Fat wedge filtrations and decomposition of polyhedral products