Milnor’s link invariants attached to certain Galois groups over \(\mathbb Q\). (English) Zbl 0990.11068

Summary: This is a résumé of the author’s recent work on certain analogies between primes and links. The purpose of this article is to introduce a new invariant, called Milnor invariant, in algebraic number theory, based on an analogy between the structure of a certain Galois group over the rational number field and that of the group of a link in three-dimensional Euclidean space. It then turns out that the Legendre, Rédei symbols are interpreted as our link invariants. We expect that this is the tip of an arithmetical theory related to the model of link theory which may give a new insight in algebraic number theory. The details will be published elsewhere.


11R32 Galois theory
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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[1] Chen, K., Fox, R. H. and Lyndon, R. C.: Free differential calculus, IV. The quotient groups of the lower central series, Ann. of Math., 68 , 81-95 (1958). · Zbl 0142.22304 · doi:10.2307/1970044
[2] Hillman, J.: Alexander Ideals of Links. Lec. Note in Math., 895 , Springer, Berlin-Heidelberg-New York (1981). · Zbl 0491.57001
[3] Hillman, J.: Algebraic invariants of links, enlarged edition of [H1] (preprint). · Zbl 1253.57001
[4] Ihara, Y.: On Galois representations arising from tower of coverings of \({\mathbf P}^{1}\backslash\{0,1,\infty\}\), Invent. Math., 86 , 427-459 (1986). · Zbl 0585.14020 · doi:10.1007/BF01389262
[5] Koch, H.: Galoissche Theorie der \(p\)-Erweiterungen. Springer, Berlin-Heidelberg-New York (1970).
[6] Milnor, J.: Link groups. Ann. of Math., 59 , 177-195 (1954). · Zbl 0055.16901 · doi:10.2307/1969685
[7] Milnor, J.: Isotopy of Links, in Algebraic Geometry and Topology, A symposium in Honour of S. Lefshetz. (eds. Fox, R. H., Spencer D. S. and Tucker, W.). Princeton Univ. Press, Princeton, pp. 280-306 (1957). · Zbl 0080.16901
[8] Murasugi, K.: Nilpotent coverings of links and Milnor’s invariant. Low-dimensional topology, London Math. Soc. Lecture Note Ser., 95 , Cambridge Univ. Press, Cambridge-New York, 106-142 (1985). · Zbl 0579.57004 · doi:10.1017/CBO9780511662744.004
[9] Murasugi, K., and Kurpita, B. I.: A Study of Braids. Math. Appl., 484 , Kluwer Acad. Publ., Dordrecht-Boston-London (1999). · Zbl 0938.57001
[10] Rédei, L.: Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper. I. J. Reine Angew. Math., 180 , 1-43 (1938). · Zbl 0021.00701 · doi:10.1515/crll.1939.180.1
[11] Reznikov, A.: Three-manifolds class field theory (Homology of coverings for a nonvirtually \(b_{1}\)-positive manifold). Sel. Math. New Ser., 3 , 361-399 (1997). · Zbl 0892.57012 · doi:10.1007/s000290050015
[12] Turaev, V. G.: Milnor’s invariants and the Massey products. J. Soviet Math., 12 , 128-137 (1979) (English transl). · Zbl 0406.55017 · doi:10.1007/BF01098424
[13] Waldspurger, J.-L.: Entrelacements sur \(\Spec(\BBZ)\). Bull. Sci. Math., 100 , 113-139 (1976). · Zbl 0343.14001
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