Higher triple Massey products and symbols
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 (in: [\textit{J. Mináč} and \textit{N. D. Tân}, J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)]. The paper under review continues the study of higher 3MP, started by the author in an earlier paper (see [\textit{E. Matzri}, J. Algebra 499, 272--280 (2018; Zbl 1425.12005)], for a proof of the vanishing, for \(p > 2\), of any defined 3MP of weight \((1, n, 1)\)). In the present paper, the author uses Pfister forms (in case \(p = 2\), see Proposition~2.1 in: [\textit{R. Elman} and \textit{T. Y. Lam}, J. Algebra 23, 181--213 (1972; Zbl 0246.15029)] and norm varieties (when \(p > 2\), see [\textit{A. Suslin} and \textit{S. Joukhovitski}, J. Pure Appl. Algebra 206, No. 1--2, 245--276 (2006; Zbl 1091.19002)]; and [\textit{C. Haesemeyer} and \textit{C. A. Weibel}, Norm varieties and the chain lemma (after Markus Rost), manuscript, available at: \url{http://www.math.uiuc.edu/K-theory /0900.}] for studying higher 3MP, where some slots are assumed to be symbols. Note that an element \(\alpha \in H^n(F)\) is called a symbol, if \(\alpha = a_1\cdot a_2 \cdots a_n\), where \(a_i \in H^1(F)\), \(i = 1, \ldots, n\). The paper under review presents two main results. The first one states that any defined 3MP of weight \((n, 1, m)\) vanishes, provided the first and last slots are symbols; the second one states that any defined 3MP of weight \((1, n, 1)\) vanishes if the middle slot is a symbol. The proof of the former theorem is obtained by reduction to the special case where \(F\) is a field of characteristic zero.
scientific article; zbMATH DE number 7041655
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