Ordinary and twisted K-theory - Clemente, Gabriel Longatto (2022)
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.