Models for the Cohomology of Certain Polyhedral Products

For a commutative ring $\Bbbk $ with unit, we describe and study various differential graded $\Bbbk $-modules and $\Bbbk $-algebras as 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, one that does not come from a geometric setting. As an application, this work sets the stage for studying the based loop space of $\Sigma (\underline {CX\!}\,,\underline {X\!}\,)^K$.