Representing 3-manifolds by triangulations of S3: a constructive approach
In a paper of I. V. Izmestʹev and M. Joswig [Adv. Geom. 3 (2003), no. 2, 191–225;], it was shown that any closed orientable 3-manifold M arises as a branched covering over S3 from some triangulation of S3. The proof of this result is based on the fact that any closed orientable 3-manifold M is a sim...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2005 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/50760 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/50760 |
| Access Level: | acceso abierto |
| Palabra clave: | 515.16 3-manifolds Topología 1210 Topología |
| Sumario: | In a paper of I. V. Izmestʹev and M. Joswig [Adv. Geom. 3 (2003), no. 2, 191–225;], it was shown that any closed orientable 3-manifold M arises as a branched covering over S3 from some triangulation of S3. The proof of this result is based on the fact that any closed orientable 3-manifold M is a simple 3-branched covering over S3 with a knot K as branched set [H. M. Hilden, Amer. J. Math. 98 (1976), no. 4, 989–997; J. M. Montesinos, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;]. In the paper under review the authors obtain the same result in a different way, which turns out to be constructive. More precisely, a triangulation Δ of the 3-sphere S3 defines uniquely a number m≤4, a subgraph Γ of Δ and a representation ω(Δ) of π1(S3∖Γ) into the symmetric group of m indices Σm. The aim of the paper is to prove that if (K,ω) is a knot or a link K in S3 together with a transitive representation ω:π1(S3∖K)→Σm, 2≤m≤3, then there is a constructive procedure to obtain a triangulation Δ of S3 such that ω(Δ)=ω. This new method involves a new representation of knots and links, called a butterfly representation. |
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