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Master's Dissertation
DOI
10.11606/D.45.2017.tde-17112017-130250
Document
Author
Full name
Luciana Basualdo Bonatto
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2017
Supervisor
Committee
Struchiner, Ivan (President)
Ferreira, Carlos Henrique Grossi
Silveira, Mariana Rodrigues da
Title in English
Bott's periodicity theorem from the algebraic topology viewpoint
Keywords in English
Botts periodicity theorem
Eilenberg-MacLane spaces
Generalised cohomology
Homotopy theory
K-Theory
Morse theory
Morse-Bott theory
Spectral sequences
Abstract in English
In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory.
Title in Portuguese
O teorema da periodicidade de Bott sob o olhar da topologia algébrica
Keywords in Portuguese
Cohomologia generalizada
Espaços de Eilenberg-MacLane
K-Teoria
Sequências espectrais
Teorema da periodicidade de Bott
Teoria de homotopia
Teoria de Morse
Teoria de Morse-Bott
Abstract in Portuguese
Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.
 
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Publishing Date
2017-12-05
 
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