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Topologies on groups related to the theory of shape and generalized coverings
Topologías en grupos asociados con la teoría de la forma y recubridores generalizados

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Sánchez González, Álvaro (2016) Topologies on groups related to the theory of shape and generalized coverings. [Thesis]

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Abstract

El presente trabajo consiste en dos partes diferenciadas: la principal de ellas (Cap tulos 1 y 2) est a dedicada a introducir estructura adicional en grupos que aparecen de manera natural en el contexto de la teor a de la forma. En la segunda parte (Cap tulo 3), se plantea c omo generalizar la teor a de espacios recubridores y, en particular, se propone una l nea de trabajo relacionada con la teor a de la forma. El punto de partida de esta tesis doctoral son los trabajos [25, 26, 68, 69, 70] en los que los autores introducen y utilizan algunas ultram etricas en el conjunto de los mor smos shape entre dos espacios topol ogicos punteados. En particular, si el dominio es (S1; 1); la construcci on realizada en [68] permite explicitar una ultram etrica en el grupo shape 1(X; x0) de un espacio m etrico compacto X; como ya fue observado en [69] y [80]. Si el espacio no es m etrico compacto, la construcci on nos lleva a utilizar el concepto de ultram etrica generalizada, en el sentido de Priess-Crampe y Ribenboim [78, 79]. En [7], D. K. Biss introduce la idea de topologizar el grupo fundamental de un espacio, de forma que la topolog a en 1(X; x0) sea una topolog a de grupo que permita detectar la (no) existencia de un recubridor universal para X: La forma de proceder sugerida es tomar en 1(X; x0)la toplog a cociente inducida por la topolog a compacto-abierta en el espacio de lazos (X; x0): Sin embargo, hay algunos errores en el art culo mencionado: en concreto, el error relacionado con el presente trabajo fue puesto de mani esto por P. Fabel en [33], mostrando que, en general, la operaci on de grupo en 1(X; x0)con la topolog a cociente no es continua. Utilizando un punto de vista similar, varios autores han tratado de dotar al grupo fundamental con una topolog a, de forma que 1(X; x0) sea un grupo topol ogico y la proyecci on q (X; x0){u100000} 1(X; x0)sea continua...

Resumen (otros idiomas)

The present work consists of two di erent parts: the main one (Chapters 1 and 2) is devoted to introduce additional structure on groups which arise naturally in the theory of shape. In the second part (Chapter 3), some generalizations of the theory of covering spaces are studied and, in particular, one is proposed according to the spirit of the theory of shape. The starting point of this doctoral thesis are the works [25, 26, 68, 69, 70] in which the authors introduced and exploited some ultrametrics in the set of shape morphisms between two (pointed) topological spaces. In particular, if the domain space is particularized in (S1; 1); the construction made in [68] allows to give an ultrametric on the shape group 1(X; x0) of a compact metric space X; as it was observed in [69] and detailed [80]. If the space X is non-compact metric, the construction leads to a generalized ultrametric, in the sense of Priess-Crampe and Ribenboim [78, 79]. In [7], D. K. Biss introduced the idea of topologizing the fundamental group of a topological space, in such a way that the topology on 1(X; x0) was a group topology which allows to detect the (non) existence of universal covering for X: The approach consists just in taking on 1(X; x0)the quotient topology from the compact-open topology on the loop space of X; (X; x0): However, there are some errors in the referred paper, speci cally, the error related with our work is revealed by P. Fabel in [33] showing that, in general, the group operation on 1(X; x0) with the quotient topology is not continuous. Using a similar point of view, di erent authors tried to endow the fundamental group with a topology such that 1(X; x0) is a topological group and the projection q (X; x0){u100000} 1(X; x0)is continuous. The idea of introducing a topology on 1(X; x0)is not new at all. Earliest works around this idea seem to be by W. Hurewicz [46] and specially by J. Dugundji [27]. After the paper of Biss, some other works have appeared in which the fundamental group is endowed with di erent topologies. The most relevant are, among others, [36] where the so-called whisker topology is used, [19, 20] where the lasso topology is introduced, and [13] where the author works with a slight modi cation of the quotient topology...

Item Type:Thesis
Additional Information:

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología, leída el 20-01-2016

Directors:
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Alonso Morón, Manuel
Giraldo Carbajo, Antonio
Uncontrolled Keywords:Geometría
Palabras clave (otros idiomas):Geometry
Subjects:Sciences > Mathematics > Geometry
ID Code:38786
Deposited On:16 Aug 2016 08:48
Last Modified:12 Dec 2018 15:12

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