All three-manifolds are pullbacks of a branched covering 𝑆³ to 𝑆³

Hilden, Hugh M.; Lozano, María Teresa; Montesinos, José María

doi:10.1090/S0002-9947-1983-0709580-4

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.

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