Journal of the European Mathematical Society

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Volume 19, Issue 1, 2017, pp. 255–284
DOI: 10.4171/JEMS/665

Published online: 2016-12-12

Triple Massey products and Galois theory

Ján Mináč[1] and Nguyên Duy Tân[2]

(1) Western University, London, Canada
(2) Western University, London, Canada

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.

Keywords: Massey products, Galois theory, unipotent representations, Zassenhaus filtration

Mináč Ján, Tân Nguyên Duy: Triple Massey products and Galois theory. J. Eur. Math. Soc. 19 (2017), 255-284. doi: 10.4171/JEMS/665