Triple Massey products and absolute Galois groups
type: Item
instance of: scholarly article
title: Triple Massey products and absolute Galois groups
zbMATH Open document ID: 1425.12004
DOI: 10.4171/JEMS/748
author: Ido Efrat
author: Eliyahu Matzri
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2017-12-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1412.7265
review text: The topic of the paper under review is incorporated in the search for new criteria that allow to distinguish the absolute Galois groups (abbr., AGGs) \(\mathcal{G}_F := \mathcal{G}(F_{\text{sep}}/F)\) of fields \(F\) from arbitrary profinite groups (AGGs are viewed as with respect to the Krull topology. General restrictions of this kind are rare: one of them is the non-existence of elements of \(\mathcal{G}_F\) of finite orders greater than \(2\), established by the Artin-Schreier theory. Another one, less explicit, is provided by the Whaples theorem, via Pontrjagin's duality theory: the restriction concerns the structure of the (continuous) character group \(X(\mathcal{G}_F)\) of \(\mathcal{G}_F\). Specifically, the Whaples theorem shows (see [Duke Math. J. 24, 201--204 (1957; Zbl 0081.26703)]) that if \(X(\mathcal{G}_F) \ne \{0\}\), then \(X(\mathcal{G}_F)\) possesses a nontrivial divisible subgroup or it is a group of exponent (period) \(2\). The theorem can be restated in terms of Galois cohomology, by identifying \(X(\mathcal{G}_F)\) with the continuous cohomology group \(H^1(\mathcal{G}_F, \mathbb Q/\mathbb Z)\); here \(\mathbb Q\) is the additive group of rational numbers, \(\mathbb Z\) is the subgroup of integers, and \(\mathbb Q/\mathbb Z\) is viewed as a discrete abelian group. The significance of Galois cohomology for the study of AGGs has deeper been understood after the proof of the Bloch-Kato Conjecture due to Voevodsky and Rost (see \textit{V. Voevodsky}'s articles in [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028); Ann. Math. (2) 174, No. 1, 401--438 (2011; Zbl 1236.14026)]. In particular, this shows that the cohomology (graded) ring \(H^{\ast}(\mathcal{G}_F) H^(\mathcal{G}_F, \mathbb Z/m)\) is generated by its degree \(1\) elements, and its relations originate from the degree \(2\) component. As shown by \textit{S. K. Chebolu} et al., this can be used to rule out many more profinite groups from being AGGs (see [Math. Ann. 35…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 16K50
zbMATH DE Number: 6821383
zbMATH Keywords: triple Massey products
zbMATH Keywords: absolute Galois groups
zbMATH Keywords: Galois cohomology
zbMATH Keywords: essential triple Massey products
MaRDI profile type: Publication
arXiv ID: 1412.7265
cites work: Q5777227
cites work: Quotients of absolute Galois groups which determine the entire Galois cohomology
cites work: Real homotopy theory of Kähler manifolds
cites work: Q3963168
cites work: The Zassenhaus filtration, Massey products, and representations of profinite groups
cites work: Vanishing of Massey Products and Brauer Groups
cites work: On the descending central sequence of absolute Galois groups
cites work: Galois groups and cohomological functors
cites work: Higher Massey products in the cohomology of mild pro-\(p\)-groups.
cites work: Algebraic invariants of links
cites work: Splitting varieties for triple Massey products
cites work: Complex geometry. An introduction
cites work: Massey Higher Products
cites work: Triple Massey products of weight \((1,n,1)\) in Galois cohomology
cites work: Triple Massey products over global fields
cites work: The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields
cites work: Triple Massey products vanish over all fields
cites work: Triple Massey products and Galois theory
cites work: Construction of unipotent Galois extensions and Massey products
cites work: Milnor invariants and Massey products for prime numbers
cites work: Q3445917
cites work: Mixed Artin-Tate motives with finite coefficients
cites work: Koszulity of cohomology \( = K(\pi,1)\)-Ness + quasi-formality
cites work: Division algebras of exponent 2 and characteristic 2
cites work: Massey products and ideal class groups
cites work: Q3861564
cites work: Corps a Involution Neutralises par une Extension Abelienne Elementaire
cites work: Motivic cohomology with \(\mathbb Z/2\)-coefficients
cites work: On motivic cohomology with \(\mathbb{Z}/l\)-coefficients
cites work: Q4681832
cites work: On the Galois group of 2-extensions with restricted ramiﬁcation
cites work: On 3-Nilpotent Obstructions to π1 Sections for $$ \mathbb{P}^{1}_\mathbb{Q}$$−{0,1, $$\infty$$}
cites work: n-Nilpotent Obstructions to pi_1 Sections of P^1-{0,1,infty} and Massey Products
OpenAlex ID: W2963836579
Recommended article: Quotients of absolute Galois groups which determine the entire Galois cohomology
Recommended article: Triple Massey products and Galois theory
Recommended article: Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture
Recommended article: Four-fold Massey products in Galois cohomology
Recommended article: The Kummerian property and maximal pro-\(p\) Galois groups
Recommended article: Homogeneous spaces, algebraic \(K\)-theory and cohomological dimension of fields
Recommended article: Relations in the maximal pro-𝑝 quotients of absolute Galois groups
Recommended article: Galois module structure of square power classes for biquadratic extensions
Recommended article: Free products of absolute Galois groups
Recommended article: From Field Theoretic to Abstract Co-Galois Theory
description: scientific article