Representations of links in relation with branched double coverings. (Spanish: Representaciones de enlaces en relación con recubridores dobles ramificados)

Abstract

Let L⊆S3 be a link and let t:L→S3 be its 2-fold branched cyclic cover. A represented link (L,ω) is a link together with a transitive representation of π=π1(S3−L) in Sn, the symmetric group of order n!. In his thesis ["On branched covers over a knot and the Poincare conjecture'' (Spanish), Ph.D. Thesis, Univ. Complutense de Madrid, Madrid, 1971] the author describes certain geometric moves (D) and conjectures: (1) Given (L,ω) it is possible to find a new link (L∗,ω∗) by a sequence of moves (D) such that L∗ splits into (L1,ω1),⋯,(Lr,ωr) and the ωi are representations to S2. This can actually be weakened as can be seen from a paper by R. H. Fox [Rev. Mat. Hisp.-Amer. (4) 32 (1972), 158–166;]. For a long time the author has also wondered whether (2) π1(L)≠1 if L is not trivial. Together, (1) and (2) imply the Poincaré conjecture. The paper under review is a study of (2): if π has a transitive representation ω:π→Sn (n≥3) so that for every meridian m of L, ω(m) is of order two then ω defines a transitive representation ψ:π/⟨m2⟩→Sn. From an observation of Fox [op. cit.] π1(L) is always a subgroup of π/⟨m2⟩ of order two. Thus, since ψ is transitive, its image must have order >2 and π1(L)≠1. In this way (2) is changed into a question of existence of transitive representations ω with ω(m)2=1. Given ω, let p:M(L,ω)→S3 be the corresponding branched cover of S3. Actually (1) can be weakened to links where π1(M)=1. In this paper the author first shows that if Fg is the oriented surface of genus g, S1×Fg is of the form M(L,ω) for representations ω:π→S4 with image the Klein group. More generally such Kleinian representations yield maps g:M(L,ω)→L which are ordinary double coverings and p=tg. Next the author studies dihedral representations ω:π→Dr (which exist if and only if H1(L) has an r-cyclic representation φ). The author defines two transitive representations α:Dr→Sr and β:Dr→S2r; let γ:π1(L)→H1(L) be abelianization. If M=M(L,αω) and N=M(L,βω) then N is a 2-fold branched cover of M and an r-fold covering space of L corresponding to the kernel of φγ:π1(L)→Zr. {There is a misprint on line 1, p. 154, where ω is a representation of π on Dp. It is unfortunate that the author uses so many similar Fraktur characters; they make the reading difficult and that accounts for the misprint}.