Degenerate fourfold Massey products over arbitrary fields
We prove that, for all fields F of characteristic different from 2 and all a,b,c in F*, the mod 2 Massey product <a,b,c,a> vanishes as soon as it is defined. For every field E of characteristic different form 2, we construct a field F containing E and a,b,c,d in F* 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 DGA of the absolute Galois group is not formal. This is joint work with Alexander Merkurjev.