Models for the Cohomology of Certain Polyhedral Products

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 $(\underset{‾}{CX},\underset{‾}{X}{)}^K$. Along the way, we prove that the integral cohomology $H^{\ast }((D^1,S^0{)}^K;\mathbb{Z})$ 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 $H^{\ast }((D^1,S^0{)}^K;\mathbb{Z})$ and in $H^{\ast }((D^n,S^{n-1}{)}^K;\mathbb{Z})$ for $n\ge 2$ have the same origin. As an application, this work sets the stage for studying the based loop space of $(\underset{‾}{CX},\underset{‾}{X}{)}^K$ in terms of the bar construction applied to the differential graded $\mathbb{Z}$-algebras $B({\mathcal{C}}^{\ast }(\underset{‾}{X};\mathbb{Z}),K)$ quasi-isomorphic to the singular cochain algebra ${\mathcal{C}}^{\ast }((\underset{‾}{CX},\underset{‾}{X}{)}^K;\mathbb{Z})$.