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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Idèlic class field theory for 3-manifolds and very admissible links


Authors: Hirofumi Niibo and Jun Ueki
Journal: Trans. Amer. Math. Soc. 371 (2019), 8467-8488
MSC (2010): Primary 57M12, 11R37; Secondary 57M99
DOI: https://doi.org/10.1090/tran/7480
Published electronically: February 28, 2019
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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.


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Additional Information

Hirofumi Niibo
Affiliation: Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
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
Email: uekijun46@gmail.com

DOI: https://doi.org/10.1090/tran/7480
Keywords: Id\`ele, class field theory, 3-manifold, branched covering, arithmetic topology.
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).
Article copyright: © Copyright 2019 American Mathematical Society