Triple Massey products and absolute Galois groups
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. 352, No. 1, 205--221 (2012; Zbl 1272.12015); Trans. Am. Math. Soc. 369, 2697--2720 (2017; Zbl 1390.12004)]). The search for new distinctive properties of AGGs has recently concentrated on triple Massey products, i.e. the special case where \(n = 3\) of \(n\)-fold Massey products \(H^1(\mathcal{G}_F)^n \to H^2(\mathcal{G}_F)\), defined as a certain multi-valued map, which coincides for \(n = 2\) with the cup product. The \(n\)-fold Massey product \(\langle \chi_1, \dots , \chi_n\rangle \in H^2(\mathcal{G}_F)\) is called essential, if it is nonempty but does not contain the zero element of \(H^2(\mathcal{G}_F)\). A series of very recent results (the authors refer to papers by Hopkins and Wickelgren, Mináč and Tân) show, that under various assumptions, the triple Massey product for \(H^{\ast }(\mathcal{G}_F)\) is never essential. Thus profinite groups \(G\) for which \(H^{\ast }(G)\) contains an essential triple Massey product cannot be realized as AGGs of fields satisfying these assumptions. By the main result of the paper under review (stated as Theorem 0.1), the triple Massey product for \(H^{\ast }(\mathcal{G}_F)\) is never essential in case \(F\) contains a primitive \(p\)-th root of unity. The authors note in an addendum that: an initial version of their proof is contained in \textit{E. Matzri}'s preprint [``Triple Massey products in Galois cohomology'', Preprint, \url{arXiv:1411.4146}]; a Galois-cohomological proof of Theorem 0.1 has also been given by \textit{J. Mináč} and \textit{N. D. Tân} [J. Lond. Math. Soc., II. Ser. 94, No. 3, 909--932 (2016; Zbl 1378.12002)], who point out that a standard restriction-corestriction argument allows one to remove the condition that \(F\) contains a primitive \(p\)-th root of unity.
scientific article