A representation of closed orientable 3-manifolds as 3-fold branched coverings of S3

Abstract

In 1920, J. W. Alexander proved that, if M3 is a closed orientable three-dimensional manifold, then there exists a covering M3→S3 that branches over a link [same Bull. 26 (1919/20), 370–372; Jbuch 47, 529]. In the paper under review, the author proves a precision, piquant reformulation of Alexander's result: M3 is a closed orientable three-manifold, then there is a threefold irregular covering M3→S3 that branches over a knot; exactly two points of M3 cover each point of the singular set (the branching knot), one point with index of branching one; the other, with index of branching two. H. M. Hilden has independently proved the same theorem [ibid. 80 (1974), 1243–1244]. Suppose that g is the genus of M3, let both Xg and Xg′ denote a handlebody of genus g, and let φ:∂Xg→∂Xg′ be a homeomorphism for which Xg∪φXg′ is a Heegard splitting of M3. Let B and B′ both denote three-cells, and let A be a collection of g+2 disjoint arcs properly imbedded in B; let A′ be a similar collection of arcs in B′. Hilden constructs two irregular three-fold coverings p:Xg→B and p′:Xg′→B′; the covering p branches over A and the covering p′, over A′. The homeomorphism φ:∂Xg→∂Xg′ (or a homeomorphism isotopic to φ) projects to a homeomorphism γ:∂B→∂B′ such that γ(A∩∂B)=A′∩∂B′ and such that A∪γ|(A∩∂B)A′ is a knot in B∪γB′, the three-sphere. The branched covering we are seeking is p∪p′:Xg∪φXg′→B∪γB′. The author proves the theorem differently. Let L denote two unliked trivial knots, K1 and K2, in S3, and let Σ3 denote the symmetric group on {0,1,2}. The assignment of a meridian of Ki to the transposition (0i) (i=1,2) induces a representation π1(S3−L)→Σ3 and, thereby, a three-fold irregular covering, p:Σ3→S3, branched over L. The manifold Σ3 is S3, and p−1(Ki) contains exactly two curves, one with branching index one, the other, K˜i, with branching index two (i=1,2). Furthermore, the curves of p−1(Ki) are unknotted and unlinked. Now surgery on an appropriate μ-link L in Σ3 produces the manifold M3 [W. B. R. Lickorish, Ann. of Math. (2) 82 (1965), 414–420]. We can assume that each component of L cuts K˜1∪K˜2 in exactly two points, and we can find a second-regular neighborhood Vj for each component kj of L such that p(Vj) is a three-cell and such that p(Vj)∩L consists of two disjoint arcs (j=1,⋯,μ). Appropriate surgery on the solid tori V1,⋯,Vμ in Σ3 induces surgery on the corresponding three-cells p(V1),⋯,p(Vμ), and one obtains a three-fold, irregular covering M3→S3, branched over a link. Then, applying tools he developed in a previous paper [Rev. Mat. Hisp-Amer. (4) 32 (1972), 33–51], the author modifies the covering so that branching occurs over a knot.