The π-geography problem and the Hurwitz problem
Let d ≥ 2 be an integer and a partition of d. In this article we study the problem of for which pairs of integers (a, b) there is a branched coating F: ∑ → D2 = {z ∈ C: | z | 6 ≤ 1} that has critical values, x (∑) = −b, and such that the monodromy that is obtained when crossing the border of D2 in a...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2009 |
| País: | Colombia |
| Institución: | Universidad EAFIT |
| Repositorio: | Repositorio EAFIT |
| Idioma: | español |
| OAI Identifier: | oai:repository.eafit.edu.co:10784/14512 |
| Acceso en línea: | http://hdl.handle.net/10784/14512 |
| Access Level: | acceso abierto |
| Palabra clave: | Branched Coating Critical Value Characteristic Of Euler Riemann – Hurwitz Formula Hurwitz Problem Monodromia Recubrimiento Ramificado Valor Crítico Característica De Euler Fórmula De Riemann–Hurwitz Monodromía |
| Sumario: | Let d ≥ 2 be an integer and a partition of d. In this article we study the problem of for which pairs of integers (a, b) there is a branched coating F: ∑ → D2 = {z ∈ C: | z | 6 ≤ 1} that has critical values, x (∑) = −b, and such that the monodromy that is obtained when crossing the border of D2 in a positive sense belongs to the conjugation class in the symmetric group Sd determined by the π partition. Four variants of this problem are studied: i) without requiring domain connection, ii) requiring domain connection, iii) without requiring domain connection, but requiring that the coating be semi-stable, iv) requiring that the domain be related and that the coating is semi-stable. Complete solutions of the first two variants are obtained, and a partial solution of the remaining variants is obtained. It also explains how the interest in these problems arises from the study of an analogous question for functions whose domain is 4-dimensional. |
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