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Four-fold Massey products in Galois cohomology

  • Pierre Guillot (a1), Ján Mináč (a2) and Adam Topaz (a3)


In this paper, we develop a new necessary and sufficient condition for the vanishing of $4$ -Massey products of elements in the modulo- $2$ Galois cohomology of a field. This new description allows us to define a splitting variety for $4$ -Massey products, which is shown in the appendix to satisfy a local-to-global principle over number fields. As a consequence, we prove that, for a number field, all such $4$ -Massey products vanish whenever they are defined. This provides new explicit restrictions on the structure of absolute Galois groups of number fields.



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[Ara75] Arason, J. K., Cohomologische Invarianten quadratischer Formen , J. Algebra 36 (1975), 448491.
[Col92] Colliot-Thélène, J.-L., L’arithmétique des variétés rationnelles , Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), 295336.
[CS07] Colliot-Thélène, J.-L. and Sansuc, J.-J., The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group) , in Algebraic groups and homogeneous spaces, Tata Institute of Fundamental Research Studies in Mathematics (Tata Institute of Fundamental Research, Mumbai, 2007), 113186.
[CS94] Colliot-Thélène, J.-L. and Swinnerton-Dyer, P., Hasse principle and weak approximation for pencils of Severi–Brauer and similar varieties , J. Reine Angew. Math. 453 (1994), 49112.
[DGMS75] Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds , Invent. Math. 29 (1975), 245274.
[Dwy75] Dwyer, W. G., Homology, Massey products and maps between groups , J. Pure Appl. Algebra 6 (1975), 177190.
[Efr14] Efrat, I., The Zassenhaus filtration, Massey products, and representations of profinite groups , Adv. Math. 263 (2014), 389411.
[EM15] Efrat, I. and Matzri, E., Vanishing of Massey products and Brauer groups , Canad. Math. Bull. 58 (2015), 730740.
[EM17] Efrat, I. and Matzri, E., Triple Massey products and absolute Galois groups , J. Eur. Math. Soc. (JEMS) 19 (2017), 36293640.
[EM11] Efrat, I. and Mináč, J., On the descending central sequence of absolute Galois groups , Amer. J. Math. 133 (2011), 15031532.
[EL76] Elman, R. and Lam, T. Y., Quadratic forms under algebraic extensions , Math. Ann. 219 (1976), 2142.
[Fen83] Fenn, R. A., Techniques of geometric topology, London Mathematical Society Lecture Note Series, vol. 57 (Cambridge University Press, Cambridge, 1983).
[GLMS03] Gao, W., Leep, D. B., Mináč, J. and Smith, T. L., Galois groups over nonrigid fields , in Valuation theory and its applications, Vol. II, Saskatoon, SK, 1999, Fields Institute Communications, vol. 33 (American Mathematical Society, Providence, RI, 2003), 6177.
[GHS03] Graber, T., Harris, J. and Starr, J., Families of rationally connected varieties , J. Amer. Math. Soc. 16 (2003), 5767.
[Gro68] Grothendieck, A., Le groupe de Brauer, I, II, III , in Dix exposés sur la cohomologie des schémas (North-Holland, Amsterdam, 1968), 46188.
[HW09] Haesemeyer, C. and Weibel, C., Norm varieties and the chain lemma (after Markus Rost) , in Algebraic topology, Abel Symposia, vol. 4 (Springer, Berlin, 2009), 95130.
[HW16] Harpaz, Y. and Wittenberg, O., On the fibration method for zero-cycles and rational points , Ann. of Math. (2) 183 (2016), 229295.
[HW15] Hopkins, M. J. and Wickelgren, K. G., Splitting varieties for triple Massey products , J. Pure Appl. Algebra 219 (2015), 13041319.
[Isa15] Isaksen, D. C., When is a fourfold Massey product defined? Proc. Amer. Math. Soc. 143 (2015), 22352239.
[Mas58] Massey, W. S., Some higher order cohomology operations , in Symposium internacional de topología algebraica International symposium on algebraic topology (Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958), 145154.
[Mat11] Matzri, E., Triple Massey products in Galois cohomology, Preprint (2014), arXiv:1411.4146.
[Mil80] Milne, J. S., Étale cohomology, Princeton Mathematical Series, vol. 33 (Princeton University Press, Princeton, NJ, 1980).
[Min96] Mináč, J. and Spira., M., Witt rings and Galois groups , Ann. of Math. (2) 144 (1996), 3560.
[MT15a] Mináč, J. and Tân, N. D., The kernel unipotent conjecture and the vanishing of Massey products for odd rigid fields , Adv. Math. 273 (2015), 242270.
[MT15b] Mináč, J. and Tân, N. D., Triple Massey products over global fields , Doc. Math. 20 (2015), 14671480.
[MT16] Mináč, J. and Tân, N. D., Triple Massey products vanish over all fields , J. Lond. Math. Soc. (2) 94 (2016), 909932.
[MT17a] Mináč, J. and Tân, N. D., Construction of unipotent Galois extensions and Massey products , Adv. Math. 304 (2017), 10211054.
[MT17b] Mináč, J. and Tân, N. D., Counting Galois U4(F p )-extensions using Massey products , J. Number Theory 176 (2017), 76112.
[MT17c] Mináč, J. and Tân, N. D., Triple Massey products and Galois theory , J. Eur. Math. Soc. (JEMS) 19 (2017), 255284.
[Mor78] Morgan, J. W., The algebraic topology of smooth algebraic varieties , Publ. Math. Inst. Hautes Études Sci. (1978), 137204.
[Mor86] Morgan, J. W., Correction to: ‘The algebraic topology of smooth algebraic varieties’ [Inst. Hautes Études Sci. Publ. Math. No. 48 (1978), 137–204; MR0516917 (80e:55020)] , Publ. Math. Inst. Hautes Études Sci. (1986), 185.
[Mor02] Morishita, M., On certain analogies between knots and primes , J. Reine Angew. Math. 550 (2002), 141167.
[Mor04] Morishita, M., Milnor invariants and Massey products for prime numbers , Compos. Math. 140 (2004), 6983.
[Neu99] Neukirch, J., Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322 (Springer, Berlin, 1999).
[NSW08] Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, second edition (Springer, Berlin, 2008).
[Ros98] Rost, M., Chain lemma for splitting fields of symbols, Preprint (1998).
[Sal85] Saltman, D. J., The Brauer group and the center of generic matrices , J. Algebra 97 (1985), 5367.
[Sha07] Sharifi, R. T., Massey products and ideal class groups , J. Reine Angew. Math. 603 (2007), 133.
[Sko96] Skorobogatov, A. N., Descent on fibrations over the projective line , Amer. J. Math. 118 (1996), 905923.
[Sul77] Sullivan, D., Infinitesimal computations in topology , Publ. Math. Inst. Hautes Études Sci. 47 (1977), 269331.
[SJ06] Suslin, A. and Joukhovitski, S., Norm varieties , J. Pure Appl. Algebra 206 (2006), 245276.
[Tat76] Tate, J., Relations between K 2 and Galois cohomology , Invent. Math. 36 (1976), 257274.
[Voe11] Voevodsky, V., On motivic cohomology with Z/l-coefficients , Ann. of Math. (2) 174 (2011), 401438.
[Vog05] Vogel, D., On the Galois group of 2-extensions with restricted ramification , J. Reine Angew. Math. 581 (2005), 117150.
[Wei09] Weibel, C., The norm residue isomorphism theorem , J. Topol. 2 (2009), 346372.
[Wic09] Wickelgren, K., Lower central series obstructions to homotopy sections of curves over number fields, PhD thesis, Stanford University (2009).
[Wic12] Wickelgren, K., n-nilpotent obstructions to 𝜋1 sections of ℙ1 -{0, 1, } and Massey products , in Galois–Teichmüller theory and arithmetic geometry, Advanced Studies in Pure Mathematics, vol. 63 (Mathematical Society of Japan, Tokyo, 2012), 579600.
[Wic12] Wickelgren, K., On 3-nilpotent obstructions to 𝜋1 sections for ℙ 1 -{0, 1, } , in The arithmetic of fundamental groups—PIA 2010, Contributions in Mathematical and Computational Sciences, vol. 2 (Springer, Heidelberg, 2012), 281328.
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Four-fold Massey products in Galois cohomology

  • Pierre Guillot (a1), Ján Mináč (a2) and Adam Topaz (a3)


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