A note on a theorem of Alexander

Abstract

represented link ("nudo coloreado'') (L,ωn) is a tame link L in S3 together with a transitive representation ωn of π1(S3−L,∗) into the symmetric group Sn; it can easily be pictured in the plane by a regular knot projection and a labelling of the overpasses by elements of Sn. With (L,ωn) there is canonically associated an n-fold covering space C(L,ωn) of S3 branched over L. J. W. Alexander showed in 1920 that any connected orientable closed 3-manifold M is C(L,ωn) for some represented link; in addition he asserted that we may have branching index ≤2 everywhere [see R. H. Fox, Topology of 3-manifolds and related topics (Proc. Univ. Georgia Inst., 1961), pp. 213–216, Prentice-Hall, Englewood Cliffs, N.J., 1962; MR0140116 (25 #3539)]. In the present paper a new proof of this assertion is given in a slightly stronger version; in fact for given M the author exhibits an n∈N and a represented link (L˜,ω˜n) such that (i) M=C(L˜,ω˜n) and moreover (ii) the label of each overpass is a permutation of (1,⋯,n) which only permutes some pair of these numbers. The method of proof consists in applying to some (L,ωn) satisfying (i) a series of modifications which leave the topological type of the covering space invariant but gradually change (L,ωn) to satisfy (ii) in the end. The usefulness of this method is illustrated by determining the topological type of some C(L,ωn); in particular, the simply connected 3-manifolds proposed by R. H. Fox [op. cit.] as possible counterexamples to the Poincaré conjecture are shown to be S3.