Toral Rank Conjecture for Moment-Angle Complexes

We consider an operation $K\mapsto L(K)$ on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as $K$-powers). The main property of the doubling operation is that the moment-angle complex $\mathscr Z_K$ can be identified with the real moment-angle complex $\mathbb R\mathscr Z_{L(K)}$ for the double $L(K)$. By way of application, we prove the toral rank conjecture for the spaces $\mathscr{Z}_K$ by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes $\mathbb R\mathscr Z_K$. This paper can be viewed as a continuation of the author's previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.