Stanley–Reisner rings of generalized truncation polytopes and their moment–angle manifolds

We consider simple polytopes $P=\mathrm{vc}^k(\Delta^{n_1}\times\dots\times\Delta^{n_r})$ for $n_1\ge\dots\ge n_r\ge1$, $r\ge1$, and $k\ge0$, that is, $k$-vertex cuts of a product of simplices, and call them generalized truncation polytopes. For these polytopes we describe the cohomology ring of the corresponding moment–angle manifold $\mathcal Z_P$ and explore some topological consequences of this calculation. We also examine minimal non-Golodness for their Stanley–Reisner rings and relate it to the property of $\mathcal Z_P$ being a connected sum of sphere products.