URL: http://eprints.ucm.es/17151Colección: Archivo institucional e-prints complutense Autor: Chumillas Checa, Valerio; Montesinos Amilibia, José María CDU: 515.162 Departamento: Fac. de CC. Matemáticas - Depto. de Geometría y Topología Editorial: Springer estatus: Publicado formato: Electrónico Identificador OAI: oai:www.ucm.es:17151 ISSN: 0025-5831 materia2: Topología Materia: Matemáticas Palabras clave: branched over homology n-spheres; irregular dihedral branched covers; cyclic branched coverings resumen> H. M. Hilden [Bull. Amer. Math. Soc. 80 (1974), 1243–1244; MR0350719 (50 #3211)], U. Hirsch [Math. Z. 140 (1974), 203–230] and Montesinos [Bull. Amer. Math. Soc. 80 (1974), 845–846] showed that every closed and orientable 3-manifold is a 3-fold dihedral covering space branched along a knot in S3. The purpose of the paper under review is to answer the question of whether this is true for irregular dihedral covering spaces branched over S3 with more than 3 sheets.
The authors first show that for each odd prime p, the homology group Hi(M;Z) of every p-fold irregular dihedral covering space M over a homology n-sphere can be given the structure of a finitely generated module over the ring Z[ξ+ξ−1] of integers of the real cyclotomic field Q[ξ+ξ−1], where ξ=exp(2Ï€i/p). For each odd prime p, using the fact that Z[ξ+ξ−1] is a Dedekind domain, they describe the class Dp of finitely generated abelian groups supporting the structure of a finitely generated module over Z[ξ+ξ−1], and prove that if M is a p-fold irregular dihedral covering space branched over a homology n-sphere, then Hi(M;Z)∈Dp, iÉ 0,n. This generalizes the results of Chumillas ["Study of dihedral coverings in S3 branched over knots'', Ph.D. Thesis, Madrid, 1984; per bibl.] and of A. Costa and J. M. Ruiz [Math. Ann. 275 (1986), no. 1, 163–168]. As a consequence of these results, they obtain 3-manifolds which are not p-fold irregular dihedral covering spaces branched over S3 for any prime p: H. M. Hilden [Bull. Amer. Math. Soc. 80 (1974), 1243–1244; MR0350719 (50 #3211)], U. Hirsch [Math. Z. 140 (1974), 203–230] and Montesinos [Bull. Amer. Math. Soc. 80 (1974), 845–846] showed that every closed and orientable 3-manifold is a 3-fold dihedral covering space branched along a knot in S3. The purpose of the paper under review is to answer the question of whether this is true for irregular dihedral covering spaces branched over S3 with more than 3 sheets.
The authors first show that for each odd prime p, the homology group Hi(M;Z) of every p-fold irregular dihedral covering space M over a homology n-sphere can be given the structure of a finitely generated module over the ring Z[ξ+ξ−1] of integers of the real cyclotomic field Q[ξ+ξ−1], where ξ=exp(2Ï€i/p). For each odd prime p, using the fact that Z[ξ+ξ−1] is a Dedekind domain, they describe the class Dp of finitely generated abelian groups supporting the structure of a finitely generated module over Z[ξ+ξ−1], and prove that if M is a p-fold irregular dihedral covering space branched over a homology n-sphere, then Hi(M;Z)∈Dp, iÉ 0,n. This generalizes the results of Chumillas ["Study of dihedral coverings in S3 branched over knots'', Ph.D. Thesis, Madrid, 1984; per bibl.] and of A. Costa and J. M. Ruiz [Math. Ann. 275 (1986), no. 1, 163–168]. As a consequence of these results, they obtain 3-manifolds which are not p-fold irregular dihedral covering spaces branched over S3 for any prime p revisado: TRUE Tipo: Artículo Título: The homology of cyclic and irregular dihedral coverings branched over homology spheres Título de publicación: Mathematische Annalen
Colección: Archivo institucional e-prints complutense