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A note on 3-fold branched coverings of S3

Published online by Cambridge University Press:  24 October 2008

José María Montesinos
Affiliation:
Universidad de Zaragoza, Spain

Extract

H. Hilden(3, 4) and the author(9,10) proved independently the following result:

Theorem 1. Each closed, orientable 3-manifold is a 3-fold, dihedral covering of S3, branched over a knot.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Fox, R. H. Some problems in knot theory. Topology of 3-manifolds, ed. Fort, M. K., Prentice Hall, 1962.Google Scholar
(2)Hempel, J. Construction of orientable 3-manifolds. Topology of 3-manifolds, ed. Fort, M. K., Prentice Hall, 1962.Google Scholar
(3)Hilden, H.Every closed, orientable 3-manifold is a 3-fold branched covering space of S 3. Bull. Amer. Math. Soc. 80 (1974), 1243–4.CrossRefGoogle Scholar
(4)Hilden, H.Three-fold branched coverings of S 3. Amer. J. Math. 98 (1976), 989997.CrossRefGoogle Scholar
(5)Hilden, H. and Montesinos, J.A method of constructing 3-manifolds and its application to the computation of the μ-invariant. Proc. of Symposia in Pure Math. A.M.S. Publications 32 (1978), 5169.Google Scholar
(6)Kirby, R.A calculus for framed links in S 3. Inventions Math. 45 (1978), 3556.CrossRefGoogle Scholar
(7)Lickorish, W. B.R. A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76 (1962), 531540.CrossRefGoogle Scholar
(8)Lickorish, W. B. R.A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769–78.CrossRefGoogle Scholar
Corrigendum Lickorish, W. B. R.A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 62 (1966), 679–81.CrossRefGoogle Scholar
(9)Montesinos, J.A representation of closed, orientable 3-manifolds as 3-fold branched coverings of S 3. Bull. Amer. Math. Soc. 80 (1974), 845–6.CrossRefGoogle Scholar
(10)Montesinos, J.Three-manifolds as 3-fold branched covers of S 3. Quarterly J. Math. Oxford (2), 27 (1976), 8594.CrossRefGoogle Scholar
(11)Montesinos, J.4-manifolds, 3-fold covering spaces and ribbons. Trans. Amer. Math. Soc. 245 (1978), 453467.Google Scholar
(12)Wallace, A. D.Modifications and cobounding manifolds. Canad. J. Math. 12 (1960), 503528.CrossRefGoogle Scholar
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