Massey products, toric topology and combinatorics of polytopes

Abstract

In this paper we introduce a direct family of simple polytopes $P^0\subset P^1\subset \dots$ such that for any $k$, $2\le k\le n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds ${\mathcal{Z}}_{P^n}$. We prove that the direct sequence of manifolds $\ast \subset S^3\hookrightarrow \dots \hookrightarrow {\mathcal{Z}}_{P^n}\hookrightarrow {\mathcal{Z}}_{P^{n+1}}\hookrightarrow \dots$ 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 $H^{\ast }(\Omega X)$ and $H^{\ast }(X)$ with coefficients in a field, where $X={\mathcal{Z}}_{P^n}$.

53 pages, 5 figures; extended version of a paper to appear in Izv. Math