Triple Massey products and Galois theory

TL;DR

This paper proves that all defined triple Massey products over any field contain zero for prime 2, extending previous results and revealing new restrictions on Galois group structures, with implications for the Bloch-Kato conjecture.

Contribution

It extends the vanishing of triple Massey products to all fields and develops new methods to analyze relations in Galois groups, advancing understanding of their structure.

Findings

Triple Massey products with prime 2 always contain zero over any field.

All Massey products of order ≥3 vanish for Demushkin groups.

New restrictions on relations in maximal pro-2 Galois groups.

Abstract

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any $n$-Massey product for some prime $p$ has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro-$p$-groups, where we do not require that all cup products vanish. We prove that all $n$-Massey products, $n\geq 3$, vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch-Kato conjecture.