A moment angle complex whose rank of double cohomology is 6 (Q2683771)

The moment-angle complex \(\mathcal Z_K\) of a simplicial complex \(K\) is a certain cellular space, which plays a key role in toric topology. Its ordinary cohomology \(H^*(\mathcal Z_K;\Bbbk)\) with coefficients in any PID \(\Bbbk\) was computed in the works by Baskakov, Buchstaber, Franz and Panov and turned out to be isomorphic to the cohomology of the Koszul complex \(\Lambda[u_1,\ldots,u_m]\otimes_{\Bbbk}\Bbbk[K]\) with differential \(d\), where \(f_0(K)=m\), the face ring \(\Bbbk[K]\) is generated by elements \(v_1,\ldots,v_m\) of degree 2 and \(d(u_i)=v_i, d(v_i)=0\). In the recent paper by Limonchenko, Panov, Song, and Stanley [3] a new differential \(d'\) was defined on the Koszul complex of a simplicial complex \(K\) and it was proved that \(d\) and \(d'\) anti-commute. This gives rise to a bicomplex and the double cohomology were defined in that paper as \(HH^*(\mathcal Z_K;\Bbbk):=H^*[H^*(\mathcal Z_K;\Bbbk),d']\). After getting a number of general results and performing various computations with the double cohomology, they stated a problem: given an even integer \(r\), construct an example of a simplicial complex \(K\) s.t. the rank of \(HH^*(\mathcal Z_K)\) is equal to \(r\). The paper under review solves this problem for \(r=6\). It turns out that the desired example is provided by a certain simplicial complex \(K\) with 8 vertices, considered previously in the paper by \textit{G. Denham} and \textit{A. I. Suciu} [Pure Appl. Math. Q. 3, No. 1, 25--60 (2007; Zbl 1169.13013)]. The author provides us with the set of minimal non-faces for \(K\) and proves the main result by considering the spectral sequence of the bicomplex mentioned above.