All three-manifolds are pullbacks of a branched covering $S^{3}$ to $S^{3}$
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- by Hugh M. Hilden, María Teresa Lozano and José María Montesinos PDF
- Trans. Amer. Math. Soc. 279 (1983), 729-735 Request permission
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
- 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
- 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, DOI 10.2140/pjm.1978.78.139
- Hugh M. Hilden and José M. Montesinos, A method of constructing $3$-manifolds and its application to the computation of the $\mu$-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
- 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
- José María Montesinos, A note on $3$-fold branched coverings of $S^{3}$, Math. Proc. Cambridge Philos. Soc. 88 (1980), no. 2, 321–325. MR 578276, DOI 10.1017/S0305004100057625
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 279 (1983), 729-735
- MSC: Primary 57N10
- DOI: https://doi.org/10.1090/S0002-9947-1983-0709580-4
- MathSciNet review: 709580