Continuous deformations of Fredholm operators in B(H) - Dias, Rodrigo Lima (2021)
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.