Twisted Borel K-theory and isomorphisms between differential models of K-theory

Abstract

In this thesis we discuss some topics about twisted K-theory calculations and equivalences between a couple of differential extension models. We start with a mathematical review of models for twisted K-theory, differential extensions for the untwisted case, and Serre spectral sequences, in order to provide an explicit link between the differential extension of the untwisted and twisted cases, in addition to giving tools for the subsequent exploration of the twists that will be used. In the first part of this thesis we determine a formula up to group extensions for the twisted K-theory for a fiber bundle over the circle S^1 with fiber a compact manifold with respect to certain twists constructed from elements of the second cohomology group of the fiber. Later this case is generalized by allowing the base to be the classifying space of a finitely generated free group and the twisting will be given by a derivation of line bundles associated to the group and the fiber. This is accompanied by examples and we finally develop a spectral sequence where the previous formulas are framed. In the second part of the thesis, a topological equivalence is developed for the Freed-Lott and Carey-Mickelsson-Wang differential extension models. Additionally, we indicate a way to achieve the differential equivalence.