Models for the Cohomology of Certain Polyhedral Products

TL;DR

This paper develops algebraic models for the cohomology of polyhedral products, revealing new structural insights and connections to loop space analysis, with implications for understanding complex topological spaces.

Contribution

It introduces differential graded algebra models for polyhedral product cohomology and links cup product structures to algebraic origins, advancing the algebraic understanding of these spaces.

Findings

Integral cohomology of real moment-angle complexes is a Tor module.

Cup product structures in different polyhedral products share the same origin.

Models facilitate studying loop spaces via bar constructions.

Abstract

For a commutative ring $\mathbf k$ with unit, we describe and study various differential graded $\mathbf k$-modules and $ \mathbf k$-algebras which are models for the cohomology of polyhedral products $(\underline{CX},\underline X)^K$. Along the way, we prove that the integral cohomology $H^*((D^1, S^0)^K; \mathbb Z)$ of the real moment-angle complex is a Tor module, the one that does not come from a geometric setting. We also reveal that the apriori different cup product structures in $H^*((D^1, S^0)^K;\mathbb Z)$ and in $H^*((D^n, S^{n-1})^K; \mathbb Z)$ for $n\geq 2$ have the same origin. As an application, this work sets the stage for studying the based loop space of $(\underline{CX}, \underline X)^K$ in terms of the bar construction applied to the differential graded $\mathbb Z$-algebras $B(\mathcal C^*(\underline X; \mathbb Z), K) $ quasi-isomorphic to the singular cochain algebra $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$.