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3-variétes qui ne sont pas des revêtements cycliques ramifiés sur S3


Montesinos Amilibia, José María (1975) 3-variétes qui ne sont pas des revêtements cycliques ramifiés sur S3. Proceedings of the American Mathematical Society, 47 . pp. 495-500. ISSN 0002-9939

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Let M denote a p-fold, branched, cyclic, covering space of S3, and suppose that the three-dimensional Smith conjecture is true for p-periodic autohomeomorphisms of S3. J. S. Birman and H. M. Hilden have constructed an algorithm for deciding whether M is homeomorphic to S3 [Bull. Amer. Math. Soc. 79 (1973), 1006–1010]. Now every closed, orientable three-manifold is a three-fold covering space of S3 branched over a knot [Hilden, ibid. 80 (1974), 1243–1244], but, in the present paper, the author shows that, if Fg is a closed, orientable surface of genus g≥1, then Fg×S1 is not a p-fold, branched cyclic covering space of S3 for any p. As the author points out, this was previously known for p=2 [R. H. Fox, Mat. Hisp.-Amer. (4) 32 (1972), 158–166; the author, Bol. Soc. Mat. Mexicana (2) 18 (1973), 1–32].

Item Type:Article
Uncontrolled Keywords:Cyclic branched covering spaces, three manifolds, three-sphere, two manifolds
Subjects:Sciences > Mathematics > Topology
ID Code:17298

J. S. Birman and H. M. Hilden, The homeomorphism problem for S3, Bull. Amer. Math. Soc. 79 (1973), 1006-1010.

R. H. Fox, A note on branched cyclic coverings of spheres, Rev. Mat. Hisp.-Amer. 32 (1972), 158-166.

J. M. Montesinos, Una familia infinita de nudos representados no separables, Rev. Mat. Hisp.-Amer. 33 (1973), 32-35.

J. M. Montesinos, Variedades de Seifert que son recubridores cíclicos ramificados de dos hojas, Bol. Soc. Mat. Mexicana 18 (1973), 1-32.

M. Newman, Integral matrices, Academic Press, New York, 1972.

E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966.

B. L. van der Waerden, Modern algebra, Vol. I, Springer, Berlin, 1930-1931; English transl., Ungar, New York, 1949.

F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308-333; ibid. 4 (1967), 87-117.

Deposited On:03 Dec 2012 10:17
Last Modified:07 Feb 2014 09:45

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