A spectral sequence for polyhedral products

TL;DR

This paper develops a spectral sequence framework to analyze the structure of polyhedral products and smash products, providing homological decompositions and insights into their ring structures, with applications to moment-angle complexes.

Contribution

It introduces a natural filtration and spectral sequence for polyhedral products, enabling detailed homological analysis and ring structure applications.

Findings

Spectral sequence for polyhedral products derived.

Homological decomposition of polyhedral smash products achieved.

Applications to ring structures of moment-angle complexes.

Abstract

The purpose of this paper is to exhibit fine structure for polyhedral products Z(K;(X,A) and polyhedral smash products $\widehat{Z}(K;(X,A)$. (Moment-angle complexes are special cases for which (X,A) = (D^2,S^1)). There are three main parts. The first defines a natural filtration of the polyhedral product and derives properties of the resulting spectral sequence. This is followed with applications. The second part uses the first to give a homological decomposition of the polyhedral smash product. Finally there are applications to the ring structure of H*(Z(K;(X,A))) for CW-pairs (X,A) satisfying suitable freeness conditions.