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

In [6], the authors construct a cochain complex C\!H^*(ZK) on the cohomology of a moment-angle complex ZK and call the resulting cohomology the double cohomology, H\!H^*(ZK). In this paper, we study the change of rank in double cohomology after gluing an n-simplex to a simplicial complex 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 K such that rank H\!H^*(ZK)=r.