The homology of simplicial complements and the cohomology of polyhedral products (Q2353625)

The moment-angle complexes have been studied by topologists for many years. \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)] introduced toric manifolds which were being studied intensively by algebraic geometers. They observed that every quasi-toric manifold is the quotient of a moment-angle complex by the free action of a real torus. The moment-angle complex is denoted by \(\mathcal{Z}_K\) corresponding to an abstract simplicial complex \(K\). The topology of \(\mathcal{Z}_K\) is complicated and getting more attention by topologists lately. Recently a lot of work has been done on generalizing the moment-angle complex \(\mathcal{Z}_K= \mathcal{Z}_K(D^2, S^1)\) to pairs of spaces \((\underline{X}, \underline{A})\). In the paper under review, the authors study the cohomology of the generalized moment-angle complexes \(\mathcal{Z}_K(\underline{X}, \underline{A})\) corresponding to the pairs of spaces \((\underline{X}, \underline{A})\) with inclusions \(A_i\hookrightarrow X_i\) being homotopic to constant for all \(i\). The authors define a simplicial complement \(\mathbb{P} =\{\sigma_1, \ldots, \sigma_s\}\) to be a sequence of subsets of \([m]\) and prove that the simplicial complement \(\mathbb{P}\) corresponds to an unique simplicial complex \(K_{\mathbb{P}}\) with vertices in \([m]\). Then, they define the homology algebra of a simplicial complement \(H_{i,\sigma}(\Lambda^{\ast,\ast}[\mathbb{P}],d)\) over a principal ideal domain \(\mathbf{k}\) and prove that \(H_{\ast,\ast}(\Lambda[\mathbb{P}],d)\) is isomorphic to the Tor-algebra of the corresponding face ring \(\mathbf{k}(K_{\mathbb{P}})\) by the Taylor resolution. This gives an algorithm for the cohomology algebraic structure of the moment-angle complex by the isomorphism \(H^{\ast}(\mathcal{Z}_{K_{\mathbb{P}}}, \mathbf{k})\cong\text{Tor}_{\ast,\ast}^{\mathbf{k[x]}}(\mathbf{k}(K_{\mathbb{P}}), \mathbf{k})\). As applications of the simplicial complement, the authors also give methods to compute the cohomology of the moment-angle complex corresponding to a link and star \(\text{link}_{K_{\mathbb{P}}}\sigma, \text{star}_{K_{\mathbb{P}}}\sigma\) and the cohomology modules of the polyhedral products as well.