Idèlic class field theory for 3-manifolds and very admissible links
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- by Hirofumi Niibo and Jun Ueki PDF
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Abstract:
We study a topological analogue of idèlic class field theory for 3-manifolds in the spirit of arithmetic topology. We first introduce the notion of a very admissible link $\mathcal {K}$ in a 3-manifold $M$, which plays a role analogous to the set of primes of a number field. For such a pair $(M,\mathcal {K})$, we introduce the notion of idèles and define the idèle class group. Then, getting the local class field theory for each knot in $\mathcal {K}$ together, we establish analogues of the global reciprocity law and the existence theorem of idèlic class field theory.References
- R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836
- Christopher Deninger, A note on arithmetic topology and dynamical systems, Algebraic number theory and algebraic geometry, Contemp. Math., vol. 300, Amer. Math. Soc., Providence, RI, 2002, pp. 99–114. MR 1936368, DOI 10.1090/conm/300/05144
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- Ralph H. Fox, Covering spaces with singularities, A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, N.J., 1957, pp. 243–257. MR 0123298
- Low-dimensional topology and number theory, Oberwolfach Rep. 9 (2012), no. 3, 2541–2596. Abstracts from the workshop held August 26–September 1, 2012; Organized by Paul E. Gunnells, Walter Neumann, Adam S. Sikora and Don Zagier. MR 3156734, DOI 10.4171/OWR/2012/42
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. MR 1336822
- M. M. Kapranov, Analogies between the Langlands correspondence and topological quantum field theory, Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993) Progr. Math., vol. 131, Birkhäuser Boston, Boston, MA, 1995, pp. 119–151. MR 1373001
- Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory. 2, Translations of Mathematical Monographs, vol. 240, American Mathematical Society, Providence, RI, 2011. Introduction to class field theory; Translated from the 1998 Japanese original by Masato Kuwata and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR 2817199, DOI 10.1090/mmono/240
- Barry Mazur, Remarks on the Alexander polynomial, unpublished note, http:\slash\slashwww.math.harvard.edu\slash~mazur\slashpapers\slashalexander_polynomial.pdf, 1963–64.
- Curtis T. McMullen, Knots which behave like the prime numbers, Compos. Math. 149 (2013), no. 8, 1235–1244. MR 3103063, DOI 10.1112/S0010437X13007173
- Tomoki Mihari, Cohomological Approach to Class Field Theory in Arithmetic Topology, Canadian Journal of Mathematics, 1-45. DOI:10.4153/CJM-2018-020-0
- J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327
- Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
- Edwin E. Moise, Affine structures in $3$-manifolds. VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159–170. MR 61822, DOI 10.2307/1969837
- Masanori Morishita, A theory of genera for cyclic coverings of links, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 7, 115–118. MR 1857286
- Masanori Morishita, On certain analogies between knots and primes, J. Reine Angew. Math. 550 (2002), 141–167. MR 1925911, DOI 10.1515/crll.2002.070
- Masanori Morishita, Analogies between knots and primes, 3-manifolds and number rings [translation of MR2208305], Sugaku Expositions 23 (2010), no. 1, 1–30. Sugaku expositions. MR 2605747
- Masanori Morishita, Knots and primes, Universitext, Springer, London, 2012. An introduction to arithmetic topology. MR 2905431, DOI 10.1007/978-1-4471-2158-9
- Masanori Morishita, Yu Takakura, Yuji Terashima, and Jun Ueki, On the universal deformations for $\textrm {SL}_2$-representations of knot groups, Tohoku Math. J. (2) 69 (2017), no. 1, 67–84. MR 3640015, DOI 10.2748/tmj/1493172129
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Hirofumi Niibo, Idèlic class field theory for 3-manifolds, Kyushu J. Math. 68 (2014), no. 2, 421–436. MR 3243372, DOI 10.2206/kyushujm.68.421
- Alexander Reznikov, Three-manifolds class field theory (homology of coverings for a nonvirtually $b_1$-positive manifold), Selecta Math. (N.S.) 3 (1997), no. 3, 361–399. MR 1481134, DOI 10.1007/s000290050015
- Alexander Reznikov, Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. (N.S.) 6 (2000), no. 1, 1–39. MR 1771215, DOI 10.1007/s000290050001
- Adam S. Sikora, Idelic topology, notes for personal use, unpublished note, (2000s).
- Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields, Comment. Math. Helv. 78 (2003), no. 4, 832–844. MR 2016698, DOI 10.1007/s00014-003-0781-x
- Adam S. Sikora, slides for the workshop “Low dimensional topology and number theory III”, March, 2011, Fukuoka, slides, 2011.
- Jun Ueki, On the homology of branched coverings of 3-manifolds, Nagoya Math. J. 213 (2014), 21–39. MR 3290684, DOI 10.1215/00277630-2393795
- Jun Ueki, On the Iwasawa $\mu$-invariants of branched $\mathbf {Z}_p$-covers, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 6, 67–72. MR 3508576, DOI 10.3792/pjaa.92.67
- Jun Ueki, On the Iwasawa invariants for links and Kida’s formula, Internat. J. Math. 28 (2017), no. 6, 1750035, 30. MR 3663789, DOI 10.1142/S0129167X17500355
- Jun Ueki, Chebotarev link is stably generic, preprint, arXiv:1902.06906.
Additional Information
- Hirofumi Niibo
- Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
- MR Author ID: 1092204
- Email: niibo.hirofumi@gmail.com
- Jun Ueki
- Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan
- Address at time of publication: Department of Mathematics, School of System Design and Technology, Tokyo Denki University, 5 Senju Asahi-cho, Adachi-ku, Tokyo, 120-8551, Japan
- MR Author ID: 1087285
- Email: uekijun46@gmail.com
- Received by editor(s): November 2, 2016
- Received by editor(s) in revised form: June 15, 2017, August 5, 2017, and October 14, 2017
- Published electronically: February 28, 2019
- Additional Notes: The authors were partially supported by Grant-in-Aid for JSPS Fellows (27-7102, 25-2241).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8467-8488
- MSC (2010): Primary 57M12, 11R37; Secondary 57M99
- DOI: https://doi.org/10.1090/tran/7480
- MathSciNet review: 3955553