On the cohomology of moment-angle complexes associated to Gorenstein* complexes

The main goal of this article is to study the cohomology rings and their applications of moment-angle complexes associated to Gorenstein* complexes, especially, the applications in combinatorial commutative algebra and combinatorics. First, we give a topological characterization of Gorenstein* complexes in terms of Alexander duality (as an application we give a topological proof of Stanley's Theorem). Next we give some cohomological transformation formulae of $\mathcal {Z}_{K}$, which are induced by some combinatorial operations on the Gorenstein* complex $K$, such as the connected sum operation and stellar subdivisions. We also prove that $\mathcal {Z}_{K}$ is a prime manifold whenever $K$ is a flag $2$-sphere by proving the indecomposability of their cohomology rings. Then we use these results to give the unique decomposition of the cohomology rings of moment-angle manifolds associated to simplicial $2$-spheres, and explain how to use it to detect the cohomological rigidity problem of these moment-angle manifolds.