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| Author(s): |
Rodrigo Lima Dias
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | São Paulo. |
| Institution: | Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) |
| Defense date: | 2021-07-29 |
| Examining board members: |
Severino Toscano do Rego Melo;
Ruy Exel Filho;
Elmar Ludwig Schrohe
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| Advisor: | Severino Toscano do Rego Melo |
| Abstract | |
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. (AU) | |
| FAPESP's process: | 18/21971-8 - Continuous deformations of Fredholm operators in B(H) |
| Grantee: | Rodrigo Lima Dias |
| Support Opportunities: | Scholarships in Brazil - Master |