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
HTML articles powered by AMS MathViewer

by Israel Berstein and Allan L. Edmonds PDF

Trans. Amer. Math. Soc. 247 (1979), 87-124 Request permission

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 ${P^2} \times {S^1}$.

References

James W. Alexander, Note on Riemann spaces, Bull. Amer. Math. Soc. 26 (1920), no. 8, 370–372. MR 1560318, DOI 10.1090/S0002-9904-1920-03319-7

A lemma on systems of knotted curves

9

Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369–383. MR 192475, DOI 10.1112/plms/s3-16.1.369
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 402776
Maurice Heins, Interior mapping of an orientable surface into $S^2$, Proc. Amer. Math. Soc. 2 (1951), 951–952. MR 45221, DOI 10.1090/S0002-9939-1951-0045221-4
Hugh M. Hilden, Three-fold branched coverings of $S^{3}$, Amer. J. Math. 98 (1976), no. 4, 989–997. MR 425968, DOI 10.2307/2374037
Ulrich Hirsch, Über offene Abbildungen auf die $3$-Sphäre, Math. Z. 140 (1974), 203–230 (German). MR 362313, DOI 10.1007/BF01214163
Ulrich Hirsch, Offene Abbildungen von Flächen auf die $2$-Sphäre mit minimalem Defekt, Arch. Math. (Basel) 27 (1976), no. 6, 649–656. MR 431173, DOI 10.1007/BF01224734
Ulrich Hirsch, On regular homotopy of branched coverings of the sphere, Manuscripta Math. 21 (1977), no. 3, 293–306. MR 482736, DOI 10.1007/BF01167881
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
A. Hurwitz, Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), no. 1, 1–60 (German). MR 1510692, DOI 10.1007/BF01199469
W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307–317. MR 145498, DOI 10.1017/s0305004100036926
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, DOI 10.1017/s030500410003824x
J. Lüroth, Note über Verzweigungsschnitte und Querschnitte in einer Riemann’schen Fläche, Math. Ann. 4 (1871), no. 2, 181–184 (German). MR 1509744, DOI 10.1007/BF01442591
José M. Montesinos, Three-manifolds as $3$-fold branched covers of $S^{3}$, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94. MR 394630, DOI 10.1093/qmath/27.1.85
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

Principes topologiques de la théorie des fonctions analytiques

René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
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, DOI 10.1016/S1385-7258(53)50009-5
John W. Wood, Foliations on $3$-manifolds, Ann. of Math. (2) 89 (1969), 336–358. MR 248873, DOI 10.2307/1970673