Triple Massey products and Galois theory

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 ≥ 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.

The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields

“…In [32] we show that Conjecture 1.1 is true for n = 3, p = 2, and all fields. This is the first case when the validity of this conjecture is established for all fields.…”

Triple Massey products and absolute Galois groups

“…1) p = 2 and F is a local field or a global field (Hopkins and Wickelgren [HW15]); 2) p = 2 and F is arbitrary (Mináč and Tân [MT14a]); 3) p is arbitrary and F is a local field (Mináč and Tân; follows from [MT14a,Th. 4.3] and [MT15, Th.…”

“…Thus profinite groups G for which H * (G) contains an essential triple Massey product cannot be realized as absolute Galois groups of fields satisfying these assumptions. In [MT14a] Mináč and Tân develop a method to produce such groups G, by examining their presentation by generators and relations modulo the 4th term in the pZassenhaus filtration. As a concrete example, the profinite group G on 5 generators σ 1 , .…”

Finite quotients of Galois pro- $$p$$ p groups and rigid fields

“…Moreover, part of the interest of this result lies in the fact that the proof is purely group-theoretical, and it does not rely on results form field theory, according to the group-theoretical approach to Galois theory, which consists in translating the arithmetic information in group-theoretical terms, forgetting the arithmetic background (as it is done with Bloch-Kato pro-p groups, see [12]), in order to get as much information as possible on the structure of Galois pro-p groups using tools from the theory of pro-p groups. Further, the proof makes use of the Zassenhaus filtration of pro-p groups, which is gaining increasing importance as tool for the study of Galois groups (see, e.g., [5] and [11]). …”