Direct families of polytopes with nontrivial Massey products

TL;DR

This paper constructs a sequence of smooth manifolds with nontrivial higher Massey products, using a new family of polytopes called flag nestohedra, advancing understanding of Massey products in topology.

Contribution

It introduces a new family of polytopes and a sequence of manifolds exhibiting nontrivial Massey products of increasing complexity.

Findings

Existence of manifolds with nontrivial higher Massey products.

Construction of a new family of flag nestohedra $\

Development of PDEs for the generating series of $\

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 $H^*(M)$ were known. In this work we introduce a sequence of smooth closed manifolds $\{M_{k}\}^{\infty}_{k=1}$ such that $M_{k}\hookrightarrow M_{k+1}$ is a submanifold and a retract of $M_{k+1}$ for any $k\geq 1$ and there exists a nontrivial Massey product $\langle\alpha_{1},\ldots,\alpha_{n}\rangle$ in $H^*(M_{k})$ for each $2\leq n\leq k$. The sequence $\{M_k\}^{\infty}_{k=1}$ is determined by a new family of flag nestohedra $\mathcal P_{Mas}$. We give P.D.E. for the two-parametric generating series of $\mathcal P_{Mas}$.