Higher Commutators in the Loop Space Homology of $K$-Products

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex $K$. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley–Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen–Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.