Families of minimally non-Golod complexes and their polyhedral products

We consider families of simple polytopes $P$ and simplicial complexes $K$ well-known in polytope theory and convex geometry, and show that their moment-angle complexes have some remarkable homotopy properties which depend on combinatorics of the underlying complexes and algebraic properties of their Stanley–Reisner rings. We introduce infinite series of Golod and minimally non-Golod simplicial complexes $K$ with moment-angle complexes ${\mathcal{Z}}_K$ having free integral cohomology but not homotopy equivalent to a wedge of spheres or a connected sum of products of spheres respectively. We then prove a criterion for a simplicial multiwedge and composition of complexes to be Golod and minimally non-Golod and present a class of minimally non-Golod polytopal spheres.