Massey products, toric topology and combinatorics of polytopes

TL;DR

This paper constructs a family of polytopes whose associated moment-angle manifolds exhibit non-trivial Massey products of all orders, revealing complex cohomological structures and spectral sequence differentials.

Contribution

It introduces a new family of polytopes with manifolds having non-trivial Massey products of arbitrary order, advancing understanding of their cohomological properties.

Findings

Existence of non-trivial Massey products of all orders in the cohomology of the manifolds

Sequence of manifolds forms a retract chain with inclusion properties

Presence of arbitrarily large differentials in the Eilenberg--Moore spectral sequence

Abstract

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ 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 $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ 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 $X=\mathcal Z_{P^n}$.