Abstract
In this chapter we review Gauss’ genus theory from the link-theoretic point of view. We shall see that the notion of genera is defined by using the idea analogous to the linking number. We also present, vice versa, a topological analogue of genus theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Fuluta, K.: On capitulation theorems for infinite groups. In: Primes and Knots. Contemporary Mathematics, vol. 416, pp. 41–47. Am. Math. Soc., Providence (2006). Preprint (2003)
Iyanaga, S., Tamagawa, T.: Sur la theorie du corps de classes sur le corps des nombres rationnels. J. Math. Soc. Jpn. 3, 220–227 (1951)
Kapranov, M.: Analogies between number fields and 3-manifolds. Unpublished Note (1996)
Morin, B.: Utilisation d’une cohomologie étale equivariante en topologie arithmétique. Compos. Math. 144(1), 32–60 (2008)
Morishita, M.: A theory of genera for cyclic coverings of links. Proc. Jpn. Acad. 77(7), 115–118 (2001)
Morishita, M.: On capitulation problem for 3-manifolds. In: Galois Theory and Modular Forms. Developments in Mathematics, vol. 11, pp. 305–313. Kluwer Academic, Dordrecht (2003)
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)
Reznikov, A.: Embedded incompressible surfaces and homology of ramified coverings of three-manifolds. Sel. Math. New Ser. 6, 1–39 (2000)
Reznikov, A., Moree, P.: Three-manifold subgroup growth, homology of coverings and simplicial volume. Asian J. Math. 1(4), 764–768 (1997)
Sikora, A.: Analogies between group actions on 3-manifolds and number fields. Comment. Math. Helv. 78, 832–844 (2003)
Ueki, J.: On the homology of branched coverings of 3-manifolds. Preprint (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag London Limited
About this chapter
Cite this chapter
Morishita, M. (2012). Homology Groups and Ideal Class Groups I—Genus Theory. In: Knots and Primes. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2158-9_6
Download citation
DOI: https://doi.org/10.1007/978-1-4471-2158-9_6
Publisher Name: Springer, London
Print ISBN: 978-1-4471-2157-2
Online ISBN: 978-1-4471-2158-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)