Montesinos Amilibia, José María and Chumillas Checa, Valerio
(1988)
*The homology of cyclic and irregular dihedral coverings branched over homology spheres.*
Mathematische Annalen, 280
(3).
pp. 483-500.
ISSN 0025-5831

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Official URL: http://www.springerlink.com/content/jl058w8718653m25/

## Abstract

H. M. Hilden [Bull. Amer. Math. Soc. 80 (1974), 1243–1244; MR0350719 (50 #3211)], U. Hirsch [Math. Z. 140 (1974), 203–230] and Montesinos [Bull. Amer. Math. Soc. 80 (1974), 845–846] showed that every closed and orientable 3-manifold is a 3-fold dihedral covering space branched along a knot in S3. The purpose of the paper under review is to answer the question of whether this is true for irregular dihedral covering spaces branched over S3 with more than 3 sheets.

The authors first show that for each odd prime p, the homology group Hi(M;Z) of every p-fold irregular dihedral covering space M over a homology n-sphere can be given the structure of a finitely generated module over the ring Z[ξ+ξ−1] of integers of the real cyclotomic field Q[ξ+ξ−1], where ξ=exp(2πi/p). For each odd prime p, using the fact that Z[ξ+ξ−1] is a Dedekind domain, they describe the class Dp of finitely generated abelian groups supporting the structure of a finitely generated module over Z[ξ+ξ−1], and prove that if M is a p-fold irregular dihedral covering space branched over a homology n-sphere, then Hi(M;Z)∈Dp, i≠0,n. This generalizes the results of Chumillas ["Study of dihedral coverings in S3 branched over knots'', Ph.D. Thesis, Madrid, 1984; per bibl.] and of A. Costa and J. M. Ruiz [Math. Ann. 275 (1986), no. 1, 163–168]. As a consequence of these results, they obtain 3-manifolds which are not p-fold irregular dihedral covering spaces branched over S3 for any prime p>3. The authors indicate that the method used in this paper is applicable to the case of cyclic covering spaces branched over a homology n-sphere. The realization problem (i.e., given a group G∈Dp, does there exist an irregular p-fold dihedral covering space p:M→S3 such that H1(M;Z) is isomorphic to G?) is also studied. Finally, the authors conclude by providing three tables which give the homology group of the p-fold irregular dihedral covering spaces of the knots of less than eleven crossings with more than 2 bridges for p=5,7,11.

Item Type: | Article |
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Uncontrolled Keywords: | branched over homology n-spheres; irregular dihedral branched covers; cyclic branched coverings |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 17151 |

References: | Burde, G.: On branched coverings ofS 3. Can. J. Math.23, 84–89 (1971) Chumillas, V.: Estudio de las cubiertas dihédricas deS 3 ramificadas sobre enlaces, Ph.D. Thesis, Madrid, 1984 Costa, A., Ruiz, J.M.: On the homology of metacyclic coverings. Math. Ann.275, 163–168 (1986) Fox, R.H.: Covering spaces with singularities. Lefschetz Symposium, Princeton Math. Series 12. 243–257, Princeton: Princeton Univ. Press 1957 Fox, R.H.: A note on branched cyclic coverings of spheres. Rev. Mat. Hisp. Am. IV Ser.32, 158–166 (1972) Hilden, H.M.: Every closed orientable 3-manifold is a 3-fold branched covering space ofS 3, Bull. Am. Math. Soc.80, 1243–1244 (1974) Hirsch, U.: Über offene Abbildungen auf die 3-Sphäre. Math. Z.140, 203–230 (1974) Kaplansky, I.: Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc.72, 327–340 (1952) Marcus, D.A.: Number fields. Berlin Heidelberg New York: Springer 1977 Milnor, J.: Introduction to algebraicK-theory. Ann. Math. Stud.72, (1971) Montesinos, J.M.: Representaciones de enlaces en relación con recubridores dobles ramificados. Collect. Mat.25, 145–157 (1974) Monstesinos, J.M.: A representation of closed, orientable 3-manifolds as 3-fold branched coverings ofS 3. Bull. Am. Math. Soc.80, 845–846 (1974) Montesinos, J.M.: Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo. Ph.D. Thesis, Madrid (1971) Rolfsen, D.: Knots and links. Berkeley: Publish or Perish 1976 Steinitz, E.: Rechteckige Systeme und Moduln in algebraischen Zahlkörpern. Math. Ann.71, 328–354 (1912) Val, P. del, Weber, C.: Plans' theorem for links (in prepagation) Washington, L.C.: Introduction to cyclotomic fields. Graduated Texts in Mathematics. Vol. 83. Berlin Heidelberg New York: Springer 1982 |

Deposited On: | 20 Nov 2012 12:47 |

Last Modified: | 07 Feb 2014 09:42 |

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