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] Joan S. Birman and Jerome Powell, Special representations for 3-manifolds, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), Academic Press, New York-London, 1979, pp. 23–51. MR 537723
  • [Hi] Hugh M. Hilden, Embeddings and branched covering spaces for three and four dimensional manifolds, Pacific J. Math. 78 (1978), no. 1, 139–147. MR 513289
  • [HM] Hugh M. Hilden and José M. Montesinos, A method of constructing 3-manifolds and its application to the computation of the 𝜇-invariant, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 61–69. MR 520523
  • [K] Rob Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 273–312. MR 520548
  • [M] José María Montesinos, A note on 3-fold branched coverings of 𝑆³, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 321–325. MR 578276, 10.1017/S0305004100057625

Similar Articles

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

Retrieve articles in all journals with MSC: 57N10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0709580-4
Article copyright: © Copyright 1983 American Mathematical Society