On the rank of the double cohomology of moment-angle complexes

In [6], the authors construct a cochain complex $C\text{\hspace{0.17em}⁣}{H}^{\ast }({\mathcal{Z}}_{\mathcal{K}})$ on the cohomology of a moment-angle complex ${\mathcal{Z}}_{\mathcal{K}}$ and call the resulting cohomology the double cohomology, $H\text{\hspace{0.17em}⁣}{H}^{\ast }({\mathcal{Z}}_{\mathcal{K}})$. In this paper, we study the change of rank in double cohomology after gluing an $n$-simplex to a simplicial complex $\mathcal{K}$ in certain conditions. As an application, we give a positive answer to an open problem in [6]: For any even integer $r$, there always exists a simlicial complex $\mathcal{K}$ such that $\mathrm{rank}H\text{\hspace{0.17em}⁣}{H}^{\ast }({\mathcal{Z}}_{\mathcal{K}})=r$.