Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



All three-manifolds are pullbacks of a branched covering $ S\sp{3}$ to $ S\sp{3}$

Authors: Hugh M. Hilden, María Teresa Lozano and José María Montesinos
Journal: Trans. Amer. Math. Soc. 279 (1983), 729-735
MSC: Primary 57N10
MathSciNet review: 709580
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: There are two main results in this paper. First, we show that every closed orientable $ 3$-manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in $ {S^3}$, call them $ {F_1}$ and $ {F_2}$; taking three copies of $ {S^3}$; splitting the first along $ {F_1}$, the second along $ {F_1}$ and $ {F_2}$, and the third along $ {F_2}$; and then pasting in the natural way. Second, we show that given any closed orientable $ 3$-manifold $ {M^3}$ there is a $ 3$-fold irregular branched covering space, $ p:{M^3} \to {S^3}$, such that $ p:{M^3} \to {S^3}$ is the pullback of the $ 3$-fold irregular branched covering space $ q:{S^3} \to {S^3}$ branched over a pair of unknotted unlinked circles.

References [Enhancements On Off] (What's this?)

  • [BP] J. S. Birman and J. Powell, Special representation for $ 3$-manifolds, Geometric Topology (J. C. Cantrell, ed.), Academic Press, 1979. MR 537723 (80g:57010)
  • [Hi] H. M. Hilden, Embeddings and branched covering spaces for three and four dimensional manifolds, Pacific J. Math. 78 (1978), 139-147. MR 513289 (80k:57002)
  • [HM] H. M. Hilden and J. M. Montesinos, A method of constructing $ 3$-manifolds and its application to the computation of the $ \mu $-invariant, Proc. Sympos. Pure Math., vol. 32, Part 2, Amer. Math. Soc., Providence, R.I., 1978, pp. 61-69. MR 520523 (80e:57003)
  • [K] R. Kirby, Problems in low dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1978, pp. 273-312. MR 520548 (80g:57002)
  • [M] J. M. Montesinos, A note on $ 3$-fold branched coverings of $ {S^3}$, Math. Proc. Cambridge Philos. Soc. 88 (1980), 321-325. MR 578276 (81k:57002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57N10

Retrieve articles in all journals with MSC: 57N10

Additional Information

Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society