Publicado

2007-07-01

Open 3-manifolds and branched coverings: a quick exposition

3-variedades abiertas y cubiertas ramificadas

Palabras clave:

Knot, Link, Manifold, String, Wild, Tame, Locally tame, Cantor set, Tangle, Branched covering, Colored knot, Smith conjecture, 2000 Mathematics Subject Classification. 57M12, 57M30, 57N10, 54B15. (en)
Nudo, Enlace, Variedad, Cuerda, Salvaje, Dócil, Localmente dócil, Conjunto de Cantor, Ovillo, Conjetura de Smith (es)

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Autores/as

  • José María Montesinos – Amilibia Universidad Complutense, Madrid, España

Abstract. Branched coverings relate closed, orientable 3-manifolds to links in S3, and open, orientable 3-manifolds to strings in S3 \ T, where T is a compact, totally disconnected tamely embedded subset of S3. Here we give the foundations of this last relationship. We introduce Fox theory of branched coverings and state the main theorems. We give examples to illustrate the theorems.

Las cubiertas ramificadas relacionan las 3-variedades orientables cerradas con los enlaces en S3 y las 3-variedades abiertas con las cuerdas en S3 \  T , donde T es un subconjunto compacto, totalmente desconectado y dócilmente encajado en S3. Aquí exponemos los fundamentos básicos de esta última relación. Introducimos la teoría de Fox de las cubiertas ramificadas y enunciamos los principales teoremas. Damos ejemplos que ilustran los teoremas.

Dedicado a María Teresa Lozano Imízcoz tras 27 años de fructífera colaboración

Referencias

Bing, R. H. The collected papers of R. //. Bing. American Mathematical Society, Providence, 1988. Ed. by Sukhjit Singh, Steve Armcntrout and Robert J. Davcrman, RI: . xix, 1654 p.

Brown, M. The monotone union of open n-cells is an open n-cell. Proc. Amer. Math. Soc. 12 (1961), 812-814.

Burde, G., and Zleschang, H. Knots. Walter de Gruytcr, Berlin – New York, 1985. Gruyter Studies in Mathematics, 5.

Coxeter, H. S. Symmetrical definitions for the binary polyhedral groups. American Mathematical Society, Providence, 1959.

Coxeter, H. S., and Moser, W. O. Generators and relations for discrete groups. Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete.

Fox, R. H. A remarkable simple closed curve. Ann. of Math. 50, 2 (1949), 264-265.

Fox, R. H. Algebraic Geometry and Topology. Princeton Univ. Press, Princeton, 1957, ch. Covering spaces with singularities, pp. 243-257. A Symposium in honor of S. Lefschetz.

Fox, R. H. Topology of 3-manifolds and related topics. In Construction of simply connected 3-manifolds (1962), Proc. The Univ. of Georgia Institute, pp. 213-216. Prentice-Hall, Englewood Cliffs, N.J.

Freudenthal, H. Über die enden diskreter ráume und gruppen. Comment. Math. Helv. 17 (1945), 1-38.

Hilden, H. M. Every closed orientable 3-manifold is a 3-fold branched covering space of S^3. Bull. Amer. Math. Soc. 80 (1974), 1243-1244.

Hunt, J. H. Branched coverings as uniform completions of unbranched coverings. Contemporary Mathematics 12 (1982), 141-155.

Moise, E. E. Geometric topology in dimensions 2 and 3. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics.

Montesinos-Amilibia, J. M. Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo. Tesis doctoral, Universidad Complutense, Madrid, 1971.

Montesinos-Amilibia, J. M. Anote on a theorem of Alexander. Rev. Mat. Hisp. Amer. 32, 4 (1972), 167-187.

Montesinos - Amilibia, J. M. A representation of closed orientable 3-manifolds as 3-fold branched coverings of S^3. Bull. Amer. Math. Soc. 80 (1974), 845-846.

Montesinos - Amilibia, J. M. Some aspects of the theory of branched coverings. In Proceedings of the seventh Spanish-Portuguese conference on mathematic (Sant Feliu de Guixois, 1980), pp. 109-126.

Montesinos - Amilibia, J. M. Lectures on 3-fold simple coverings and 3-manifolds. Combinatorial methods in topology and algebraic geometry. Amer. Math. Soc., 1985. Rochester, N.Y., 1982, 157-177, Contemp. Math., 44, Providence, RI,.

Montesinos - Amilibia, J. M. Representing open 3-manifolds as 3-fold branched coverings. Rev. Mat. Complut. 15, 2 (2002), 533-542.

Montesinos - Amilibia, J. M. 3-manifolds, wild subsets of S^3 and branched coverings. Rev. Mat. Complut. 16, 2 (2003), 577-600.

Montesinos - Amilibia, J. M. Embedding strings in the unknot. J. Math. Sci. Univ. Tokyo 10, 4 (2003), 631-660.

Montesinos - Amilibia, J. M. On Cantor sets in 3-manifolds and branched coverings. Q. J. Math. 54, 2 (2003), 209-212.

Montesinos - Amilibia, J. M. Uncountably many wild knots whose cyclic branched covering are S 3. Rev. Mat. Complut. 16, 1 (2003), 329-344.

Montesinos - Amilibia, J. M. Branched coverings after Fox. Bol. Soc. Mat. Mexicana 11, 1 (2005), 19-64.

Neuwirth, L. P. K not groups. Princeton University Press, Princeton, 1965. Annals of Mathematics Studies (56).

Seifert, H., and Threlfall, W. Topologie. Naturforschung und Medizin in Deutschland 1939-1946. Dieterich’sclie Verlagsbuchhandlung, Wiesbaden, 1948.

Spanier, E. H. Algebraic topology. Springer-Verlag, New York-Berlin, 1981.

Tejada, D. M. Construction of 3-manifolds and branched coverings. L ed. Mat. 18, 1 (1997), 3-2 1.

Cómo citar

APA

Montesinos – Amilibia, J. M. (2007). Open 3-manifolds and branched coverings: a quick exposition. Revista Colombiana de Matemáticas, 41(2), 287–302. Recuperado a partir de https://revistas.unal.edu.co/index.php/recolma/article/view/94730

ACM

[1]
Montesinos – Amilibia, J.M. 2007. Open 3-manifolds and branched coverings: a quick exposition. Revista Colombiana de Matemáticas. 41, 2 (jul. 2007), 287–302.

ACS

(1)
Montesinos – Amilibia, J. M. Open 3-manifolds and branched coverings: a quick exposition. rev.colomb.mat 2007, 41, 287-302.

ABNT

MONTESINOS – AMILIBIA, J. M. Open 3-manifolds and branched coverings: a quick exposition. Revista Colombiana de Matemáticas, [S. l.], v. 41, n. 2, p. 287–302, 2007. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/94730. Acesso em: 26 may. 2022.

Chicago

Montesinos – Amilibia, José María. 2007. «Open 3-manifolds and branched coverings: a quick exposition». Revista Colombiana De Matemáticas 41 (2):287-302. https://revistas.unal.edu.co/index.php/recolma/article/view/94730.

Harvard

Montesinos – Amilibia, J. M. (2007) «Open 3-manifolds and branched coverings: a quick exposition», Revista Colombiana de Matemáticas, 41(2), pp. 287–302. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94730 (Accedido: 26mayo2022).

IEEE

[1]
J. M. Montesinos – Amilibia, «Open 3-manifolds and branched coverings: a quick exposition», rev.colomb.mat, vol. 41, n.º 2, pp. 287–302, jul. 2007.

MLA

Montesinos – Amilibia, J. M. «Open 3-manifolds and branched coverings: a quick exposition». Revista Colombiana de Matemáticas, vol. 41, n.º 2, julio de 2007, pp. 287-02, https://revistas.unal.edu.co/index.php/recolma/article/view/94730.

Turabian

Montesinos – Amilibia, José María. «Open 3-manifolds and branched coverings: a quick exposition». Revista Colombiana de Matemáticas 41, no. 2 (julio 1, 2007): 287–302. Accedido mayo 26, 2022. https://revistas.unal.edu.co/index.php/recolma/article/view/94730.

Vancouver

1.
Montesinos – Amilibia JM. Open 3-manifolds and branched coverings: a quick exposition. rev.colomb.mat [Internet]. 1 de julio de 2007 [citado 26 de mayo de 2022];41(2):287-302. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94730

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