Bott\'s periodicity theorem from the algebraic topology viewpoint

• Autor(a) principal: Bonatto, Luciana Basualdo
• Data de Publicação: 2017
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: http://www.teses.usp.br/teses/disponiveis/45/45131/tde-17112017-130250/
• topic: Botts periodicity theorem; Cohomologia generalizada; Eilenberg-MacLane spaces; Espaços de Eilenberg-MacLane; Generalised cohomology; Homotopy theory; K-Teoria; K-Theory; Morse theory; Morse-Bott theory; Sequências espectrais; Spectral sequences; Teorema da periodicidade de Bott; Teoria de homotopia; Teoria de Morse; Teoria de Morse-Bott

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.