On the construction of branched coverings of low-dimensional manifolds

Berstein, Israel; Edmonds, Allan L.

doi:10.1090/S0002-9947-1979-0517687-9

On the construction of branched coverings of low-dimensional manifolds

Authors: Israel Berstein and Allan L. Edmonds
Journal: Trans. Amer. Math. Soc. 247 (1979), 87-124
MSC: Primary 57M10
DOI: https://doi.org/10.1090/S0002-9947-1979-0517687-9
MathSciNet review: 517687
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Abstract: Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of ${S^3}$ , ${S^2}\, \times \,{S^1}$ , ${P^2}\, \times \,{S^1}$ , or the nontrivial ${S^3}$ -bundle over ${S^1}$ , and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of ${P^2}\, \times \,{S^1}$ of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of ${P^2}\, \times \,{S^1}$ .

[A1] James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), no. 8, 370–372. MR 1560318, https://doi.org/10.1090/S0002-9904-1920-03319-7
[A2] -, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93-95.
[B] Armand Borel, Seminar on transformation groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. MR 0116341
[E] D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 0192475, https://doi.org/10.1112/plms/s3-16.1.369
[F] George K. Francis, Assembling compact Riemann surfaces with given boundary curves and branch points on the sphere, Illinois J. Math. 20 (1976), no. 2, 198–217. MR 0402776
[He] Maurice Heins, Interior mapping of an orientable surface into 𝑆², Proc. Amer. Math. Soc. 2 (1951), 951–952. MR 45221, https://doi.org/10.1090/S0002-9939-1951-0045221-4
[Hil] Hugh M. Hilden, Three-fold branched coverings of 𝑆³, Amer. J. Math. 98 (1976), no. 4, 989–997. MR 425968, https://doi.org/10.2307/2374037
[Hir1] Ulrich Hirsch, Über offene Abbildungen auf die 3-Sphäre, Math. Z. 140 (1974), 203–230 (German). MR 362313, https://doi.org/10.1007/BF01214163
[Hir2] Ulrich Hirsch, Offene Abbildungen von Flächen auf die 2-Sphäre mit minimalem Defekt, Arch. Math. (Basel) 27 (1976), no. 6, 649–656. MR 431173, https://doi.org/10.1007/BF01224734
[Hir3] Ulrich Hirsch, On regular homotopy of branched coverings of the sphere, Manuscripta Math. 21 (1977), no. 3, 293–306. MR 482736, https://doi.org/10.1007/BF01167881
[Hud] J. F. P. Hudson, Piecewise linear topology, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844
[Hur] A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, https://doi.org/10.1007/BF01199469
[Li1] W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 145498, https://doi.org/10.1017/s0305004100036926
[Li2] W. B. R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR 171269, https://doi.org/10.1017/s030500410003824x
[Lu] J. Lüroth, Note über Verzweigungsschnitte und Querschnitte in einer Riemann’schen Fläche, Math. Ann. 4 (1871), no. 2, 181–184 (German). MR 1509744, https://doi.org/10.1007/BF01442591
[M] José M. Montesinos, Three-manifolds as 3-fold branched covers of 𝑆³, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94. MR 394630, https://doi.org/10.1093/qmath/27.1.85
[Sta] John Stallings, On fibering certain 3-manifolds, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 95–100. MR 0158375
[Sto] S. Stöilow, Principes topologiques de la théorie des fonctions analytiques, Gauthier-Villars, Paris, 1938.
[T] René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, https://doi.org/10.1007/BF02566923
[Wi] R. J. Wille, Sur la transformation intérieure d’une surface non orientable sur le plan projectif, Nederl. Akad. Wetensch. Proc. Ser. A. 56 = Indagationes Math. 15 (1953), 63–65 (French). MR 0054059
[Wo] John W. Wood, Foliations on 3-manifolds, Ann. of Math. (2) 89 (1969), 336–358. MR 248873, https://doi.org/10.2307/1970673