Continuous deformations of Fredholm operators in B(H)

• Ano de defesa: 2021
• Autor(a) principal: Dias, Rodrigo Lima
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Fredholm index; Fredholm operators; Index theory; Índice de Fredholm; K-teoria; K-teoria de espaços compactos; K-theory; K-theory of compact spaces; Operadores de Fredholm; Teoria do índice
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-09082021-231927/

Let X be a compact Hausdorff topological space. The K-group of X, denoted by K(X), is the Grothendieck group associated to the commutative monoid of isomorphism classes of complex vector bundles over X, equipped with the Whitney sum. Let H be an infinite dimensional Hilbert space and F(H) be the set of Fredholm operators on H. The Atiyah-Jänich Theorem states that the families-index is a natural isomorphism between the monoid of homotopy classes of functions from X into F(H) and the group K(X). In case X is a singleton, the families-index is the classic Fredholm index, and the Atiyah-Jänich Theorem states that the arcwise connected components of F(H) are characterized by the Fredholm index. In this work, we give a detailed exposition of the Atiyah-Jänich Theorem, studying the necessary elements to understand the construction of the K-group of a compact Hausdorff topological space, the definition of the families-index and giving a proof that such an index is the mentioned isomorphism.