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 \((\underline{CX\negthinspace }\,,\underline{X\negthinspace }\,)^K\) . 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 \(\Sigma(\underline{CX\negthinspace }\,,\underline{X\negthinspace }\,)^K\) .