Nagoya Mathematical Journal

On the homology of branched coverings of 3-manifolds

Jun Ueki

Full-text: Open access

Abstract

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.

Article information

Source
Nagoya Math. J., Volume 213 (2014), 21-39.

Dates
First available in Project Euclid: 26 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1385480157

Digital Object Identifier
doi:10.1215/00277630-2393795

Mathematical Reviews number (MathSciNet)
MR3290684

Zentralblatt MATH identifier
1295.57002

Subjects
Primary: 57M12: Special coverings, e.g. branched
Secondary: 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50] 52M27 11R29: Class numbers, class groups, discriminants 11R32: Galois theory

Citation

Ueki, Jun. On the homology of branched coverings of 3-manifolds. Nagoya Math. J. 213 (2014), 21--39. doi:10.1215/00277630-2393795. https://projecteuclid.org/euclid.nmj/1385480157


Export citation

References

  • [B] K. S. Brown, Cohomology of Groups, Grad. Texts in Math. 87, Springer, New York, 1982.
  • [D] C. Deninger, “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.
  • [F] P. Furtwängler, Über die Klassenzahlen der Kreisteilungskörper, J. Reine Angew. Math. 140 (1911), 29–32.
  • [H] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.
  • [HMM] J. Hillman, D. Matei, and M. Morishita, “Pro-$p$ link groups and $p$-homology groups” in Primes and Knots (Baltimore, 2003), Contemp. Math. 416, Amer. Math. Soc., Providence, 2006, 121–136.
  • [I1] K. Iwasawa, A note on the group of units of an algebraic number field, J. Math. Pures Appl. (9) 35 (1956), 189–192.
  • [I2] K. Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Semin. Univ. Hambg. 20 (1956), 257–258.
  • [I3] K. Iwasawa, On $\Gamma$-extensions of algebraic number fields, Bull. Amer. Math. Soc. (N.S.) 65 (1959), 183–226.
  • [KM] T. Kadokami and Y. Mizusawa, Iwasawa type formula for covers of a link in a rational homology sphere, J. Knot Theory Ramifications 17 (2008), 1199–1221.
  • [K] M. M. Kapranov, “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.
  • [Ma] B. Mazur, Remarks on the Alexander polynomial, 1963–1964, http://www.math.harvard.edu/~mazur/papers/alexander_polynomial.pdf (accessed 3 October 2013).
  • [Mo1] M. Morishita, A theory of genera for cyclic coverings of links, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), 115–118.
  • [Mo2] M. Morishita, “On capitulation problem for 3-manifolds” in Galois Theory and Modular Forms (Tokyo, 2001), Dev. Math. 11, Kluwer, Boston, 2004, 305–313.
  • [Mo3] M. Morishita, Analogies between knots and primes, 3-manifolds and number rings, Sugaku Expositions 23 (2010), 1–30.
  • [Mo4] M. Morishita, Knots and Primes: An Introduction to Arithmetic Topology, Universitext, Springer, London, 2012.
  • [Mn1] B. Morin, Utilisation d’une cohomologie étale équivariante en topologie arithmétique, Compos. Math. 144 (2008), 32–60.
  • [Mn2] B. Morin, Sur le topos Weil-étale d’un corps de nombres, Ph.D. dissertation, Université Bordeaux 1, Talence, France, 2008.
  • [Ra] N. Ramachandran, A note on arithmetic topology, C. R. Math. Acad. Sci. Soc. R. Can. 23 (2001), 130–135.
  • [Re] A. Reznikov, Embedded incompressible surfaces and homology of ramified coverings of three-manifolds, Selecta Math. (N.S.) 6 (2000), 1–39.
  • [Se] J.-P. Serre, Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979.
  • [Si] A. Sikora, Analogies between group actions on 3-manifolds and number fields, Comment. Math. Helv. 78 (2003), 832–844.
  • [W] L. C. Washington, Introduction to Cyclotomic Fields, Grad. Texts in Math. 83, Springer, New York, 1982.
  • [Y] H. Yokoi, On the class number of a relatively cyclic number field, Nagoya Math. J. 29 (1967), 31–44.