The strong Massey vanishing conjecture for fields with virtual cohomological dimension at most $1$

Abstract

We show that a strong vanishing conjecture for $n$-fold Massey products holds for fields of virtual cohomological dimension at most $1$ using a theorem of Haran. We also prove the same for PpC fields, using results of Haran--Jarden. Finally we construct a pro-$2$ group which satisfies the weak Massey vanishing property for every $n\geq3$, but does not satisfy the strong Massey vanishing property for $n=4$.