The polyhedral product functor: a method of computation for moment-angle complexes, arrangements and related spaces

TL;DR

This paper introduces a geometric decomposition method for generalized moment-angle complexes and related spaces, leading to new insights in homology, Stanley-Reisner rings, and homotopy theory.

Contribution

It provides a natural geometric decomposition of suspension spaces arising from polyhedral products, extending known homological and homotopy results to a broader class of spaces.

Findings

Homological decompositions for complements of subspace arrangements.

Additive decompositions for Stanley-Reisner rings.

Generalizations of homotopy theoretic results.

Abstract

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\it partial product spaces} which arise as {\it polyhedral product functors} described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \cite{goresky.macpherson}, Hochster\cite{hochster}, Baskakov \cite{baskakov}, Panov \cite{panov}, and Buchstaber-Panov \cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter \cite{porter} and Ganea \cite{ganea}. The spirit of the work here follows that of Denham-Suciu in \cite{denham.suciu}.