Degenerate fourfold Massey products over arbitrary fields

Abstract

We prove that, for all fields $F$ of characteristic different from 2 and all $a,b,c\in F^{\times }$, the mod 2 Massey product $⟨a,b,c,a⟩$ vanishes as soon as it is defined. For every field $F_0$, we construct a field $F$ containing $F_0$ and $a,b,c,d\in F^{\times }$ such that $⟨a,b,c⟩$ and $⟨b,c,d⟩$ vanish but $⟨a,b,c,d⟩$ is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod 2 cochain differential graded algebra of the absolute Galois group is not formal.