Seifert manifolds that are ramified two-sheeted cyclic coverings. (Spanish)

Abstract

If L is a link in the 3-sphere S3, let e:L˜→S3 denote the 2-fold cyclic covering of S3 branched over L. R. H. Fox [Rev. Mat. Hisp.-Amer. (4) 32 (1972), 158–166;] has shown that there is no link L in S3 such that L˜ is S1×S1×S1; the author [ibid. (4) 33 (1973), 32–35] has extended this to Fg×S1 (g≥1), where Fg denotes a closed orientable surface of genus g. In the present article he investigates the following more general question: Given any orientable Seifert fibre space M, determine whether M is homeomorphic to L˜ for some link L⊂S3; if the answer is yes, describe L. He finds an affirmative answer for all orientable Seifert fibre spaces over a 2-sphere or over a nonorientable closed surface as base B. In these cases a corresponding link L is constructed by using the technique of tangle modification introduced by J. H. Conway [Computational problems in abstract algebra (Proc. Conf., Oxford, 1967), pp. 329–358, Pergamon, Oxford, 1970;], to which corresponds the operation of removing from L˜ a solid torus and sewing it back differently in the covering. For orientable base B of positive genus g, i.e., B=Fg (g≥1), the situation is more complex: (i) The author finds a negative answer to the above question for the fibre spaces (Oog|b) without exceptional fibres, provided b≠±1,±2 and g≥1 (for the notation, see H. Seifert's article [Acta Math. 60 (1933), 147–238; Zbl 6, 83]). (ii) Analyzing the special assumption that the unique nontrivial covering transformation of the 2-fold cover is fibre-preserving, the author obtains a list of Seifert fibre spaces with base Fg, each of which is homeomorphic to L˜ for an appropriate link L in S3. (iii) The verification that this list is complete would depend on an affirmative answer to an unsolved question concerning involutions in Seifert fibre spaces. (iv) Modifying the main question, the author proves that each orientable Seifert fibre space over Fg (g≥0) is a 2-fold cyclic cover branched over a link of Hg, the 3-sphere with g handles attached. Finally, it is shown how some of these results extend from the class of Seifert fibre spaces to the class of "graph-manifolds'' introduced by F. Waldhausen [Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117;]. The paper is a fine piece of geometry, being specified throughout with interesting examples.