Publication: All three-manifolds are pullbacks of a branched covering S3 to S3
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1983-10
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American Mathematical Society
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Abstract
This paper establishes two new ways of representing all closed orientable 3-manifolds. (1) Let F,N be a pair of disjoint bounded orientable surfaces in the 3-sphere S3. Let (Sk,Fk,Nk), k=1,2,3, be 3 copies of the triplet (S,F,N). Split S1 along F1; S2 along F2 and N2; S3 along N3. Glue F1 to F2, N2 to N3 to obtain a closed orientable 3-manifold. Then every closed orientable 3-manifold can be obtained in this way. (2) Let q:S→S be any 3-fold irregular branched covering of the 3-sphere S over itself. Let M be any 3-manifold. Then there is a 3-fold irregular branched covering p:M→S and a smooth map f:S→S such that f is transverse to the branch set of q and p is the pullback of q and f.
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J. S. Birman and J. Powell, Special representation for 3-manifolds, Geometric Topology (J. C. Cantrell, ed.), Academic Press, 1979.
H. M. Hilden, Embeddings and branched covering spaces for three and four dimensional manifolds, Pacific J. Math. 78 (1978), 139-147.
H. M. Hilden and J. M. Montesinos, A method of constructing 3-manifolds and its application to the computation of the μ-invariant, Proc. Sympos. Pure Math., vol. 32, Part 2, Amer. Math. Soc., Providence, R.I., 1978, pp. 61-69.
R. Kirby, Problems in low dimensional manifold theory, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1978, pp. 273-312.
J. M. Montesinos, A note on 3-fold branched coverings of S3, Math. Proc. Cambridge Philos. Soc. 88 (1980), 321-325.