A unifying view toward polyhedral products through panel structures

TL;DR

This paper introduces a unifying framework for polyhedral products over spaces with faces, enabling comprehensive analysis of their topological and algebraic properties, including cohomology and equivariant cohomology, generalizing classical face rings.

Contribution

It defines a new notion of polyhedral product over spaces with faces, unifying various constructions and deriving new results in their topological and algebraic structures.

Findings

Computed stable decompositions of these spaces

Analyzed cohomology ring structures

Introduced the topological face ring concept

Abstract

A panel structure on a topological space is just a locally finite family of closed subspaces. A space together with a panel structure is called a space with faces. In this paper, we introduce a notion of polyhedral product over a space with faces. This notion provides a unifying viewpoint on the constructions of polyhedral products and generalized moment-angle complexes in various settings. We compute the stable decomposition of these spaces and use it to study their cohomology ring structures. Moreover, we can compute the equivariant cohomology ring of the moment-angle complex over a space with faces with respect to the canonical torus action. The calculation leads to the notion of topological face ring of a space with faces, which generalizes the classical notion of face ring (Stanley-Reisner ring) of a simplicial complex. We will see that many known results in the study of polyhedral products and moment-angle complexes can be reinterpreted from our general theorems on the polyhedral product over a space with faces. Moreover, we can derive some new results via our approach in some settings.