The notion of a Massey operation in cohomology of a differential graded algebra is well-known in algebraic topology and homological algebra. Non-trivial Massey products serve as an obstruction to Golodness of a local ring in homological algebra and to formality of a space in rational homotopy theory. According to P. May, they also determine differentials in the Eilenberg-Moore spectral sequence and generate a kernel of the cohomology suspension homomorphism.

In this talk, using the theory of direct families of polytopes developed by V. Buchstaber, we introduce sequences $\{M_{n}\}^{\infty}_{n=1}$ of moment-angle manifolds over 2-truncated cubes such that for any $n\geq 2$: $M_{n}$ is a submanifold and a retract of $M_{n+1}$, and there exists a non-trivial Massey product $\langle\alpha^{n}_{1},\ldots,\alpha^{n}_{k}\rangle$ in $H^*(M_{n})$ with $\dim\alpha^{n}_{i}=3, 1\leq i\leq k$ for every $k$, $2\leq k\leq n$.

As an application of our constructions and results, we examine nontriviality of differentials in the Eilenberg-Moore and Milnor-Moore spectral sequences for $M_n$.

This talk is based on a joint work with Victor Buchstaber (Steklov Mathematical Institute, Moscow).