Higher triple Massey products and symbols
type: Item
instance of: scholarly article
title: Higher triple Massey products and symbols
zbMATH Open document ID: 1447.12002
author: Eliyahu Matzri
published in: Journal of Algebra
publication date: 2019-03-25T00:00:00Z
review text: Let \(F\) be a field,\(G_F\) its absolute Galois group, and \(p\) a prime number different from the characteristic of \(F\). Suppose that \(F\) contains a primitive \(p\)-th root of unity and denote by \(H^n\) the (continuous) cohomology group \(H^n(G_F, \mathbb{Z}/p\mathbb{Z})\). In recent years there has been increasing interest in specific external operations called \(d\)-fold Massey products on the cohomology \(\mathbb{Z}/p\mathbb{Z}\)-algebra \(H(G_F) = \oplus H^n(F)\) of the differential graded \(\mathbb{Z}/p\mathbb{Z}\) \(C(G_F) = \oplus C^n(F)\) of continuous cochains on \(G_F\). For \(d\ge 2\), the \(d\)-fold Massey product is a family, indexed by \(\mathbb{N}^d\) of certain semi-defined, multi-valued functions, which we denote by dMP, of weight \((n_1, \ldots , n_d)\) (for a precise definition, see [\textit{D. Kraines}, Trans. Am. Math. Soc. 124, 431--449 (1966; Zbl 0146.19201)]. Specifically, for \(d = 3\), the 3MP of weight \((n, k, m)\) is a semi-defined multi-valued function as above \(\langle \cdot, \cdot , \cdot \rangle \colon H^n(F) \times H^k(F) \times H^m(F) \to H^{n+k+m-1} (F)\). By a vanishing dMP, we mean a defined dMP which contains the zero element. One of the main conjectures (due to \textit{J. Mináč} and \textit{N. D. Tân}, J. Eur. Math. Soc. 19, No. 1, 255--284 (2017; Zbl 1372.12004)) in this context is that in Galois cohomology for \(d\geq 3\), every defined dMP of weight \((1, 1, \ldots, 1)\) vanishes. This conjecture (if true) can serve as a subtle obstruction for a profinite group to be realizable as an absolute Galois group of a field (for more details, see [ Mináč-Tân (loc. cit.)]. When \(d = 3\), it has been proved by the author [``Triple Massey products and Galois cohomology'', Preprint, \url{arXiv:1411.4146}] (see also [\textit{I. Efrat} and \textit{E. Matzri}, J. Eur. Math. Soc. 19, No. 12, 3629--3640 (2017; Zbl 1425.12004)], and generalized to the case where the ground field does not contain a primitive \(p\)-th root of unity (i…
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 7041655
zbMATH Keywords: Massey products
zbMATH Keywords: Galois cohomology
zbMATH Keywords: external cohomological operations
zbMATH Keywords: symbols
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.02.028
OpenAlex ID: W2921466230
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Wikidata QID: Q128294233
DOI: 10.1016/J.JALGEBRA.2019.02.028
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description: scientific article; zbMATH DE number 7041655