Triple Massey products and absolute Galois groups

Abstract

Let p be a prime number, F a field containing a root of unity of order p, and G F the absolute Galois group. Extending results of Hopkins, Wickelgren, Mináč and Tân, we prove that the triple Massey product H 1 (G F ) 3 → H 2 (G F ) contains 0 whenever it is nonempty. This gives a new restriction on the possible profinite group structure of G F .A main problem in modern Galois theory is to understand the grouptheoretic structure of absolute Galois groups G F = Gal(F sep /F ) of fields F , that is, the possible symmetry patterns of roots of polynomials. General restrictions on the possible structure of the profinite group G F are rare: By classical results of Artin and Schreier, the torsion in G F can consist only of involutions. In addition, the celebrated work of Voevodsky and Rost ([Voe03], [Voe11]) identifies the cohomology ring H * (G F ) = H * (G F , Z/m) with the mod-m Milnor K-ring K M * (F )/m, assuming existence of m-th roots of unity. In particular, the graded ring H * (G F ) is generated by its degree 1 elements, and its relations originate from the degree 2 component. This can be used to rule out many more profinite groups from being absolute Galois groups of fields ([CEM12], [EM11b]). In fact, the Artin-Schreier restriction about the torsion also follows from the latter results [EM11b, Ex. 6.4(2)].Very recently, a remarkable series of works by Hopkins, Wickelgren, Mináč and Tân indicated the possible existence of a new kind of general restrictions on the structure of absolute Galois groups, related to the differential graded algebra C * (G F ) = C * (G F , Z/m) of continuous cochains on G F . The interplay between C * (G F ) and its cohomology algebra H * (G F ) gives rise to external operations on H * (G F ), in addition to its ("internal") ring structure with respect to the cup product, notably, the n-fold Massey

Triple Massey products vanish over all fields

“…Efrat and E. Matzri posted [EMa2] on arXiv. The paper [EMa2] is a replacement of [Ma]. In [EMa2], I. Efrat and E. Matzri also provide a cohomological approach to Theorem 4.10.…”

“…The paper [EMa2] is a replacement of [Ma]. In [EMa2], I. Efrat and E. Matzri also provide a cohomological approach to Theorem 4.10. Their approach has a similar flavor to our proofs in this paper, but it is still different.…”

Construction of unipotent Galois extensions and Massey products

“…This established the Vanishing 3-Massey Conjecture in the form formulated in [MT1]. Shortly after [Ma] appeared on the arXiv, two new preprints, [EMa2] and [MT5], appeared nearly simultaneously and independently on the arXiv as well. In [EMa2], I. Efrat and E. Matzri replace [Ma] and provide a cohomological approach to the proof of the main result in [Ma].…”

“…Matzri's result says that defined triple Massey products vanish over all fields containing a primitive p-th root of unity. Alternative cohomological proofs for Matzri's result are in [EMa2] and [MT5]. Our new proof given in this section of the crucial "non-degenerate" part of this result (see Proposition 5.3), which relies on explicit constructions of U 4 (F p )-extensions, is a very natural proof because of Dwyer's result [Dwy,Theorem 2.4].…”

“…Shortly after [Ma] appeared on the arXiv, two new preprints, [EMa2] and [MT5], appeared nearly simultaneously and independently on the arXiv as well. In [EMa2], I. Efrat and E. Matzri replace [Ma] and provide a cohomological approach to the proof of the main result in [Ma]. In [MT5] we also provide a cohomological method of proving the same result.…”