Models for the Cohomology of Certain Polyhedral Products
Martin Bendersky, Jelena Grbić
TL;DR
The paper develops differential graded algebra (DGA) and Tor-based models for the cohomology of polyhedral products $(\underline{CX},\underline{X})^K$ over a ring $\mathbb{k}$, with a focus on additive decompositions and multiplicative structures. It begins by treating real moment-angle complexes, showing their integral cohomology is a Tor module without a geometric origin and establishing additive isomorphisms to Tor via explicit DGAs such as $\bar{R}(K)$ and the model $B(K)$. It then generalizes to arbitrary polyhedral products using a generalized Stanley–Reisner ring $SR(\underline{X},K)$ and complexes $C(\underline{X},K)$, proving that $H^*((\underline{CX},\underline{X})^K;\mathbb{k})$ is a direct summand of a Tor module and providing concrete additive decompositions via $\bar{R}_J(K)$. In the algebra-model section, the authors construct $B(\underline{X},K)$ and $B(\mathcal{C}^*(\underline{X}),K)$ to realize cohomology as a DGA and connect to the bar construction for loop spaces, while comparing commutativity properties and identifying when the algebra structures align (notably for suspension spaces). Overall, the work lays a robust framework linking combinatorial data of $K$ with cohomological algebra of polyhedral products and sets the stage for loop-space computations using 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 $\mathcal C^*((\underline{CX},\underline X)^K;\mathbb Z)$.