A note on 3-fold branched coverings of S3

Abstract

For any closed orientable 3-manifold M there is a framed link (L,μ) in S3 such that M is the boundary of a 4-manifold W4(L,μ) obtained by adding 2-handles to the 4-ball along components of the framed link L. A link is symmetric if it is a union of a strongly invertible link about R1⊂R2⊂R3+ and a split link of trivial components in R3+∖R2. The author shows (Theorem 2) that there is an algorithm to obtain from a given framed link in S3 a framed symmetric link that determines the same 3-manifold. A coloured ribbon manifold (M,ω) is an immersion M in S3 with only ribbon singularities of a disjoint union of disks with handles together with a function ω from the set of components of M to the set {1,2}. Such an (M,ω) determines uniquely an oriented 4-manifold V4(M,ω) as an irregular 3-fold covering of D4, as was shown by the author [Trans. Amer. Math. Soc. 245 (1978/79), 453–467;]. Theorem 3: There is an algorithm to obtain from a framed symmetric link (L,μ) a coloured ribbon manifold (M,ω) such that W4(L,μ)≈V4(M,ω). These results yield a new proof of the theorem that each closed orientable 3-manifold is a 3-fold dihedral covering of S3, branched over a knot [cf. H. M. Hilden, Amer. J. Math. 98 (1976), no. 4, 989–997; the author, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;].