Homología efectiva y sucesiones espectrales

Effective homology and spectral sequences are two different techniques of Algebraic Topology which can be used for the computation of homology and homotopy groups. In this work we try to relate both methods, showing that the effective homology method can also be used to produce algorithms computing...

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Autor principal:	Romero Ibáñez, Ana
Otros Autores:	Rubio García, Julio (Universidad de La Rioja)
Formato:	text (thesis)
Lenguaje:	spa
Publicado:	Universidad de La Rioja (España) 2007
Materias:	Effective homology and spectral sequences
Acceso en línea:	https://dialnet.unirioja.es/servlet/oaites?codigo=1379
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id	oai-TES0000000868
record_format	dspace
institution	DialNet
collection	DialNet
language	spa
topic	Effective homology and spectral sequences
spellingShingle	Effective homology and spectral sequences Romero Ibáñez, Ana Homología efectiva y sucesiones espectrales
description	Effective homology and spectral sequences are two different techniques of Algebraic Topology which can be used for the computation of homology and homotopy groups. In this work we try to relate both methods, showing that the effective homology method can also be used to produce algorithms computing some spectral sequences. In the thesis we focus our attention on two particular situations: spectral sequences associated with filtered complexes and the Bousfield-Kan spectral sequence. The first part of the memoir is devoted to spectral sequences of filtered complexes, which under some good conditions converge to the homology groups of the initial complex. Making use of the effective homology method, we have developed several algorithms computing the different components of these spectral sequences: groups and differential maps in every stage, convergence level and filtration of the homology groups induced by the initial filtration. These algorithms have been implemented as a new module for the Kenzo system, a program of Symbolic Computation in Algebraic Topology, and can be used to determine two classical examples of spectral sequences, those of Serre and Eilenberg-Moore. Other spectral sequences are not defined by means of filtered complexes. This is the case of the Bousfield-Kan spectral sequence, related with the computation of homotopy groups, which is studied in the second part of the thesis. Our algorithms for the computation of spectral sequences of filtered complexes cannot be applied, but the effective homology method can be useful again to develop a constructive version of the Bousfield-Kan spectral sequence. As a first necessary step, our main result is an algorithm computing the effective homology of the free simplicial Abelian group RX generated by a 1-reduced simplicial set X. This algorithm allows the construction of the levels 1 and 2 of the spectral sequence; for the computation of the higher "pages", we present a sketch of a new algorithm which is not finished yet. Furthermore, we include a proof of the convergence of the spectral sequence.
author2	Rubio García, Julio (Universidad de La Rioja)
author_facet	Rubio García, Julio (Universidad de La Rioja) Romero Ibáñez, Ana
format	text (thesis)
author	Romero Ibáñez, Ana
author_sort	Romero Ibáñez, Ana
title	Homología efectiva y sucesiones espectrales
title_short	Homología efectiva y sucesiones espectrales
title_full	Homología efectiva y sucesiones espectrales
title_fullStr	Homología efectiva y sucesiones espectrales
title_full_unstemmed	Homología efectiva y sucesiones espectrales
title_sort	homología efectiva y sucesiones espectrales
publisher	Universidad de La Rioja (España)
publishDate	2007
url	https://dialnet.unirioja.es/servlet/oaites?codigo=1379
work_keys_str_mv	AT romeroibanezana homologiaefectivaysucesionesespectrales
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spelling	oai-TES00000008682019-05-11Homología efectiva y sucesiones espectralesRomero Ibáñez, AnaEffective homology and spectral sequencesEffective homology and spectral sequences are two different techniques of Algebraic Topology which can be used for the computation of homology and homotopy groups. In this work we try to relate both methods, showing that the effective homology method can also be used to produce algorithms computing some spectral sequences. In the thesis we focus our attention on two particular situations: spectral sequences associated with filtered complexes and the Bousfield-Kan spectral sequence. The first part of the memoir is devoted to spectral sequences of filtered complexes, which under some good conditions converge to the homology groups of the initial complex. Making use of the effective homology method, we have developed several algorithms computing the different components of these spectral sequences: groups and differential maps in every stage, convergence level and filtration of the homology groups induced by the initial filtration. These algorithms have been implemented as a new module for the Kenzo system, a program of Symbolic Computation in Algebraic Topology, and can be used to determine two classical examples of spectral sequences, those of Serre and Eilenberg-Moore. Other spectral sequences are not defined by means of filtered complexes. This is the case of the Bousfield-Kan spectral sequence, related with the computation of homotopy groups, which is studied in the second part of the thesis. Our algorithms for the computation of spectral sequences of filtered complexes cannot be applied, but the effective homology method can be useful again to develop a constructive version of the Bousfield-Kan spectral sequence. As a first necessary step, our main result is an algorithm computing the effective homology of the free simplicial Abelian group RX generated by a 1-reduced simplicial set X. This algorithm allows the construction of the levels 1 and 2 of the spectral sequence; for the computation of the higher "pages", we present a sketch of a new algorithm which is not finished yet. Furthermore, we include a proof of the convergence of the spectral sequence.La homología efectiva y las sucesiones espectrales son dos técnicas diferentes de la Topología Algebraica que pueden ser utilizadas para el cálculo de grupos de homología y de homotopía. En este trabajo tratamos de relacionar ambos métodos, mostrando que el método de la homología efectiva puede ser utilizado también para producir algoritmos que calculen algunas sucesiones espectrales. En la tesis nos centramos en dos situaciones particulares: las sucesiones espectrales asociadas a complejos filtrados y la sucesión espectral de Bousfield-Kan. La primera parte de la tesis está dedicada a las sucesiones espectrales de complejos filtrados, que bajo ciertas condiciones convergen a los grupos de homología del complejo inicial. Utilizando el método de la homología efectiva hemos desarrollado varios algoritmos para calcular las distintas componentes de estas sucesiones espectrales: grupos y diferenciales en todos los niveles, nivel de convergencia y filtración de los grupos de homología inducida por la filtración inicial. Estos algoritmos han sido implementados como un nuevo módulo para el sistema Kenzo, un programa de Cálculo Simbólico en Topología Algebraica, y pueden ser utilizados para calcular dos de los ejemplos clásicos de sucesión espectral, las de Serre y Eilenberg-Moore. Otras sucesiones espectrales no vienen definidas por medio de ningún complejo filtrado. Es el caso de la sucesión espectral de Bousfield-Kan, relacionada con el cálculo de grupos de homotopía, que estudiamos en la segunda parte de la tesis. Nuestros algoritmos para el cálculo de sucesiones espectrales de complejos filtrados no se pueden utilizar, pero el método de la homología efectiva puede ser útil de nuevo para desarrollar una versión constructiva de la sucesión espectral de Bousfield-Kan. Como un primer paso necesario, nuestro resultado principal es un algoritmo que calcula la homología efectiva del grupo Abeliano simplicial libre RX generado por un conjunto simplicial 1-reducido X. Este algoritmo permite construir los niveles 1 y 2 de la sucesión espectral; para el cálculo del resto de "páginas", presentamos el esquema de un nuevo algoritmo que todavía no está terminado. Además, incluimos una prueba de la convergencia de la sucesión espectral.Universidad de La Rioja (España)Rubio García, Julio (Universidad de La Rioja)Sergeraert, Francis (Université Joseph-Fourier. 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Homología efectiva y sucesiones espectrales

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