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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2018.tde-26072018-164304
Document
Author
Full name
Ana Maria Mathias Morita
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2018
Supervisor
Committee
Mattos, Denise de (President)
Melo, Thiago de
Negreiros, Caio Jose Colletti
Pergher, Pedro Luiz Queiroz
 
Title in Portuguese
Existência de ações livres e o anel de cohomologia de espaços de órbitas para variedades de Dold
Keywords in Portuguese
Ação livre
Espaço de órbitas
Sequência espectral
Variedade de Dold
Abstract in Portuguese
Sejam G um grupo topológico e X um espaço topológico. Existe uma questão natural associada ao par (G; X) sobre a existência de ações livres e contínuas de G em X. Se tal ação existe, outra questão natural é o estudo de propriedades do espaço de órbitas X / G e, nesse contexto, temos o problema usualmente difícil de se calcular o anel de cohomologia de X / G. Este trabalho é dedicado a essas questões quando X são variedades de Dold P(m;n) especiais e G = Z2. A variedade fechada e suave P(m;n), de dimensão m+2n, é o espaço de órbitas da involução livre T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) e foi introduzida por Albrecht Dold em 1956, sendo bastante estudada na literatura e desempenhando papel fundamental na teoria de cobordismo. A principal ferramenta utilizada nesse estudo foi a sequência espectral de Leray-Serre associada à fibração de Borel X → XG → BG; onde XG = (X × EG) / G é a construção de Borel associada ao G-fibrado universal EG → BG.
 
Title in English
Existence of free actions and the cohomology ring of orbit spaces for Dold manifolds
Keywords in English
Dold manifold
Free action
Orbit space
Spectral sequence
Abstract in English
Let G be a topological group and X be a topological space. There is a natural question associated with the pair (G; X) about the existence of a continuous free action of G on X. If such an action exists, other natural question is the study of properties of the orbit space X / G and, in this setting, the study of the cohomology ring of X / G. This thesis is devoted to these questions when X are special Dold manifolds P(m;n) and G = Z2. The closed smooth (m+2n)-dimensional manifold, P(m;n), is the orbit space of the free involution T : Sm × CPn → Sm × CPn (x; [z]) → (-x; [ z̄ ]) and was introduced by Albrecht Dold in 1956, being well studied in literature and playing a fundamental role in cobordism theory. The main tool used in this study was the Leray-Serre spectral sequence associated with the Borel fibration X → XG → BG; where XG = (X × EG) / G is the Borel construction associated with the universal G-bundle EG → BG.
 
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Publishing Date
2018-07-26
 
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