Massey products, toric topology and combinatorics of polytopes

Abstract

In this paper we introduce a direct family of simple polytopes P0⊂P1⊂⋯ such that for any 2≤k≤n there are non-trivial strictly defined Massey products of order k in the cohomology rings of their moment-angle manifolds ZPn. We prove that the direct sequence of manifolds ∗⊂S3↪⋯↪ZPn↪ZPn+1↪⋯ has the following properties: every manifold ZPn is a retract of ZPn+1, and one has inverse sequences in cohomology (over n and k, where k→∞ as n→∞) of the Massey products constructed. As an application we get that there are non-trivial differentials dk, for arbitrarily large k as n→∞, in the Eilenberg–Moore spectral sequence connecting the rings H∗(ΩX) and H∗(X) with coefficients in a field, where X=ZPn.