Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes $P^{0} {\subset}  P^{1} {\subset}{\kern 1pt}{\cdots}$ such that for any $2 {\leq} k {\leq} n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of theirmoment-angle manifolds$\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $*\subset S^{3}\hookrightarrow…\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}} {\hookrightarrow} {\cdots}$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed.As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$, in the Eilenberg–Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$.

polyhedral product, moment-angle manifold, Massey product, Lusternik–Schnirelmann category, polytope family, flag polytope, generating series, nestohedron, graph-associahedron