On Polyhedral Product Spaces over Polyhedral Joins

TL;DR

This paper explores the cohomological structure of polyhedral product spaces over polyhedral joins, providing a generalized formula for their Hilbert-Poincaré series and extending previous constructions in algebraic topology.

Contribution

It introduces a cohomological decomposition for polyhedral products over polyhedral joins and generalizes existing formulas for the Hilbert-Poincaré series.

Findings

Cohomological decomposition formula derived for certain families

Generalized Hilbert-Poincaré series formula established

Extension of previous constructions in polyhedral topology

Abstract

The construction of a simplicial complex given by polyhedral joins (introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and Gitler's $J$-construction and simplicial wedge construction. This article gives a cohomological decomposition of a polyhedral product over a polyhedral join for certain families of pairs of simplicial complexes. A formula for the Hilbert-Poincar\'{e} series is given, which generalizes Ayzenberg's formula for the moment-angle complex.