Título: | A representation of closed orientable 3-manifolds as 3-fold branched coverings of S3 |
Autores: | Montesinos Amilibia, José María |
Tipo de documento: | texto impreso |
Editorial: | American Mathematical Society, 1974 |
Dimensiones: | application/pdf |
Nota general: | info:eu-repo/semantics/restrictedAccess |
Idiomas: | |
Palabras clave: | Estado = Publicado , Materia = Ciencias: Matemáticas: Topología , Tipo = Artículo |
Resumen: |
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 ?:?Xg??Xg? be a homeomorphism for which Xg??Xg? 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 ?:?Xg??Xg? (or a homeomorphism isotopic to ?) projects to a homeomorphism ?:?B??B? such that ?(A??B)=A???B? and such that A??|(A??B)A? is a knot in B??B?, the three-sphere. The branched covering we are seeking is p?p?:Xg??Xg??B??B?. The author proves the theorem differently. Let L denote two unliked trivial knots, K1 and K2, in S3, and let ?3 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 ?1(S3?L)??3 and, thereby, a three-fold irregular covering, p:?3?S3, branched over L. The manifold ?3 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 ?-link L in ?3 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,?,?). Appropriate surgery on the solid tori V1,?,V? in ?3 induces surgery on the corresponding three-cells p(V1),?,p(V?), 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. |
En línea: | https://eprints.ucm.es/id/eprint/17300/1/Montesinos31.pdf |
Ejemplares
Estado |
---|
ningún ejemplar |