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 Z or ⌈dimK2⌉-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.