Ordinary and twisted K-theory

• Ano de defesa: 2022
• Autor(a) principal: Clemente, Gabriel Longatto
• Orientador(a): Ruffino, Fabio Ferrari
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: K-teoria topológica; Teorias cohomológicas generalizadas; K-teoria ordinária; Geometria de spin; Isomorfismo de thom; Mapa de gysin; Operadores de Fredholm; Aplicação do índice; K-teoria torcida
• Palavras-chave em Inglês: Topological K-theory; Generalized cohomology theories; Ordinary K-theory; Spin geometry; Thom isomorphism; Gysin map; Fredholm operators; Index map; Twisted K-theory
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA::GEOMETRIA E TOPOLOGIA::TOPOLOGIA ALGEBRICA
• Link de acesso: https://repositorio.ufscar.br/handle/20.500.14289/15841

The main topic of this thesis consists in Ordinary and Twisted Topological K-Theory. We begin by describing generalized cohomology theories through the Eilenberg-Steenrod Axioms, in order to set Ordinary K-Theory in these terms. This allows us to deduce its structural properties from the framework of generalized cohomology. Then, we expose the elementary notions of Spin Geometry to relate it to Ordinary K-Theory through the Atiyah-Bott-Shapiro Theorem. This result enables us to construct the Thom isomorphism as well as the integration map, which is known as Gysin map. After that, we rephrase Ordinary K-Theory by means of the Index map, which provides an interpretation of K-Theory through homotopy classes of continuous functions. Afterwards, we deal with Twisted K-Theory. First, we introduce the Grothendieck group of twisted vector bundles as a model for finite-order Twisted K-Theory. Then, we describe the infinite-dimensional model, through suitable bundles of Fredholm operators, that holds for twisting classes of any order. Finally, we compare these two models in the finite-order setting.