# On direct families of polytopes with nontrivial Massey products

The problem of determining nontrivial Massey products in cohomology of a space is well-known in algebraic topology and homological algebra. A number of publications was devoted to establishing formality of certain classes of smooth manifolds in the sense of rational homotopy theory; existence of a nontrivial Massey product in cohomology is an obstruction to formality of a space. Until now, few examples of manifolds $M$ with nontrivial higher Massey products in $H^*(M)$ have been constructed.

In this talk, using the theory of direct families of polytopes, we introduce sequences of moment-angle manifolds over 2-truncated cubes $\{M_{k}\}^{\infty}_{k=1}$ such that for any $n\geq 1$: $M_{n}$ is a submanifold and a retract of $M_{n+1}$, and there exists a nontrivial 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 each $2\leq k\leq n$.

This talk is based on a joint work with Victor M. Buchstaber (Steklov Mathematical Insitute of RAS, Moscow).