Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-02T22:49:32.405Z Has data issue: false hasContentIssue false
  • English
  • Français

On the homology of branched coverings of 3-manifolds

Published online by Cambridge University Press:  11 January 2016

Jun Ueki
Jun Ueki*
Affiliation:
Graduate School of Mathematics Kyushu University, Fukuoka-city, Fukuoka, 819-0395, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Following the analogies between 3-manifolds and number rings in arithmetic topology, we study the homology of branched covers of 3-manifolds. In particular, we show some analogues of Iwasawa’s theorems on ideal class groups and unit groups, Hilbert’s Satz 90, and some genus-theory–type results in the context of 3-dimensional topology. We also prove that the 2-cycles valued Tate cohomology of branched Galois covers is a topological invariant, and we give a new insight into the analogy between 2-cycle groups and unit groups.

Keywords

57M1212G0552M2711R2911R32
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 213 , March 2014 , pp. 21 - 39
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[B] Brown, K. S., Cohomology of Groups, Grad. Texts in Math. 87, Springer, New York, 1982. MR 0672956.Google Scholar
[D] Deninger, C., “A note on arithmetic topology and dynamical systems” in Algebraic Number Theory and Algebraic Geometry, Contemp. Math. 300, Amer. Math. Soc., Providence, 2002, 99–114. MR 1936368. DOI 10.1090/conm/300/05144.Google Scholar
[F] Furtwängler, P., Über die Klassenzahlen der Kreisteilungskörper, J. Reine Angew. Math. 140 (1911), 29–32.Google Scholar
[H] Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002. MR 1867354.Google Scholar
[HMM] Hillman, J., Matei, D., and Morishita, M., “Pro-p link groups and p-homology groups” in Primes and Knots (Baltimore, 2003), Contemp. Math. 416, Amer. Math. Soc., Providence, 2006, 121–136. MR 2276139. DOI 10.1090/conm/416/ 07890.Google Scholar
[I1] Iwasawa, K., A note on the group of units of an algebraic number field, J. Math. Pures Appl. (9) 35 (1956), 189–192. MR 0076803.Google Scholar
[I2] Iwasawa, K., A note on class numbers of algebraic number fields, Abh. Math. Semin. Univ. Hambg. 20 (1956), 257–258. MR 0083013.Google Scholar
[I3] Iwasawa, K., On Γ-extensions of algebraic number fields, Bull. Amer. Math. Soc. (N.S.) 65 (1959), 183–226. MR 0124316.Google Scholar
[KM] Kadokami, T. and Mizusawa, Y., Iwasawa type formula for covers of a link in a rational homology sphere, J. Knot Theory Ramifications 17 (2008), 1199–1221. MR 2460171. DOI 10.1142/S0218216508006580.CrossRefGoogle Scholar
[K] Kapranov, M. M., “Analogies between the Langlands correspondence and topological quantum field theory” in Functional Analysis on the Eve of the 21st Century, Vol. 1. (New Brunswick, N.J., 1993), Progr. Math. 131, Birkhäuser, Boston, 1995, 119–151. MR 1372994.Google Scholar
[Ma] Mazur, B., Remarks on the Alexander polynomial, 1963–1964, http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf (accessed 3 October 2013 ).Google Scholar
[Mo1] Morishita, M., A theory of genera for cyclic coverings of links, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), 115–118. MR 1857286.CrossRefGoogle Scholar
[Mo2] Morishita, M., “On capitulation problem for 3-manifolds” in Galois Theory and Modular Forms (Tokyo, 2001), Dev. Math. 11, Kluwer, Boston, 2004, 305–313. MR 2059769. DOI 10.1007/978-1-4613-0249-0_15.Google Scholar
[Mo3] Morishita, M., Analogies between knots and primes, 3-manifolds and number rings, Sugaku Expositions 23 (2010), 1–30. MR 2605747.Google Scholar
[Mo4] Morishita, M., Knots and Primes: An Introduction to Arithmetic Topology, Universitext, Springer, London, 2012. MR 2905431. DOI 10.1007/978-1-4471-2158-9.Google Scholar
[Mn1] Morin, B., Utilisation d’une cohomologie étale équivariante en topologie arithmétique, Compos. Math. 144 (2008), 32–60. MR 2388555. DOI 10.1112/ S0010437X07003168.Google Scholar
[Mn2] Morin, B., Sur le topos Weil-étale d’un corps de nombres, Ph.D. dissertation, Université Bordeaux 1, Talence, France, 2008.Google Scholar
[Ra] Ramachandran, N., A note on arithmetic topology, C. R. Math. Acad. Sci. Soc. R. Can. 23 (2001), 130–135. MR 1869056.Google Scholar
[Re] Reznikov, A., Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. (N.S.) 6 (2000), 1–39. MR 1771215. DOI 10.1007/s000290050001.CrossRefGoogle Scholar
[Se] Serre, J.-P., Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979. MR 0554237.Google Scholar
[Si] Sikora, A., Analogies between group actions on 3-manifolds and number fields, Comment. Math. Helv. 78 (2003), 832–844. MR 2016698. DOI 10.1007/ s00014-003-0781-x.Google Scholar
[W] Washington, L. C., Introduction to Cyclotomic Fields, Grad. Texts in Math. 83, Springer, New York, 1982. MR 0718674. DOI 10.1007/978-1-4684-0133-2.Google Scholar
[Y] Yokoi, H., On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31–44. MR 0207681.Google Scholar