The homology of cyclic and irregular dihedral coverings branched over homology spheres

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(2πi/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, i≠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>3. The authors indicate that the method used in this paper is applicable to the case of cyclic covering spaces branched over a homology n-sphere. The realization problem (i.e., given a group G∈Dp, does there exist an irregular p-fold dihedral covering space p:M→S3 such that H1(M;Z) is isomorphic to G?) is also studied. Finally, the authors conclude by providing three tables which give the homology group of the p-fold irregular dihedral covering spaces of the knots of less than eleven crossings with more than 2 bridges for p=5,7,11.Comité Conjunto Hispano-NorteamericanoNSFDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Cuadros para una historia de la Música

Incluye material elaborado por alumnos y profesoresTrabaja las áreas de Música y de Inglés desde un punto de vista interdisciplinar para conseguir un mayor grado de competencia comunicativa e interés en dichas materias. Los objetivos son: facilitar al alumnado el dominio de todas las estructuras básicas; interesarle en la cultura de la que esa lengua es vehículo; desarrollar los fundamentos lingüísticos, sociolingüísticos, discursivos y estratégicos en los que se basa la comunicación, facilitar la articulación de determinados sonidos, a partir del conocimiento de los órganos de fonación; facilitar el reconocimiento de las diferentes cualidades del sonido y apreciar las características de los diferentes estilos, obras y autores musicales. Las actividades desarrolladas buscan la motivación, participación e integración del alumnado a partir de dramatizaciones, interpretaciones, juegos, canciones y audiciones, insistiendo en la comprensión auditiva (distintas voces, sonidos, timbres, ritmos o estados de ánimo). Evalúa el grado de consecución de los objetivos a partir de la evaluación académica y una encuesta.Madrid (Comunidad Autónoma). Consejería de Educación y CulturaMadridMadrid (Comunidad Autónoma). Subdirección General de Formación del Profesorado. CRIF Las Acacias; General Ricardos 179 - 28025 Madrid; Tel. + 34915250893ES

The homology of cyclic and irregular dihedral coverings branched over homology spheres

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(2πi/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, i≠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>3. The authors indicate that the method used in this paper is applicable to the case of cyclic covering spaces branched over a homology n-sphere. The realization problem (i.e., given a group G∈Dp, does there exist an irregular p-fold dihedral covering space p:M→S3 such that H1(M;Z) is isomorphic to G?) is also studied. Finally, the authors conclude by providing three tables which give the homology group of the p-fold irregular dihedral covering spaces of the knots of less than eleven crossings with more than 2 bridges for p=5,7,11