Direct families of polytopes with nontrivial Massey products

Abstract

The problem of existence of nontrivial Massey products in cohomology of a space is well-known in algebraic topology and homological algebra. A number of problems in complex geometry, symplectic geometry, and algebraic topology can be stated in terms of Massey products. One of such problems is to establish formality of smooth manifolds in rational homotopy theory. There have already been constructed a few classes of spaces with nontrivial triple Massey products in cohomology. Until now, very few examples of manifolds $M$ with nontrivial higher Massey products in $H^{\ast }(M)$ were known. In this work we introduce a sequence of smooth closed manifolds $\{M_k{\}}_{k=1}^{\infty }$ such that $M_k\hookrightarrow M_{k+1}$ is a submanifold and a retract of $M_{k+1}$ for any $k\ge 1$ and there exists a nontrivial Massey product $⟨{\alpha }_1,\dots ,{\alpha }_n⟩$ in $H^{\ast }(M_k)$ for each $2\le n\le k$. The sequence $\{M_k{\}}_{k=1}^{\infty }$ is determined by a new family of flag nestohedra ${\mathcal{P}}_{Mas}$. We give P.D.E. for the two-parametric generating series of ${\mathcal{P}}_{Mas}$.

28 pages, 5 figures