Lie groups and twisted K-Theory

Richardson, Samuel 2022. Lie groups and twisted K-Theory. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

In this thesis we will build to a result that has applications in C*- algebras and twisted K-theory. We aim to understand the behaviour of exponential functors and the cohomology theories their target categories induce. We will use the Weyl map, K-theory, and the suspension-loop adjunction in order to achieve this goal. We will begin by acquainting ourselves with fibre bundles and a few key theorems and definitions that we will make heavy use of later due to the role fibre bundles play in defining the 0th complex topological K-theory group. We will discuss a few important functors, adjunctions, and characteristic classes. We will also give a description of the cohomology ring of any flag manifold Fn(C k ) as a quotient ring of the polynomial ring with n generators. We will also begin to understand generalised cohomology theories. Exponential functors are a particular family of monoidal functors between strict symmetric monoidal categories. We will show that each of these functors induce a family of natural transformations and that the suspension isomorphisms from the 0th degree to the 1st degree commute with the relevant natural transformations. The source category of an exponential functor is always a category that we will call C⊕ and the cohomology theory it induces is connective K-theory. We will investigate the effect an exponential functor has on a vector bundle. We will describe the Weyl map and we will discover that the class of this map in K-theory corresponds to a sum of tensor products of certain formal differences of line bundles with circle components. Finally it will be shown that the class of the Weyl map in our more exotic cohomology theories corresponds to a very similar class where we have instead taken a formal quotient of vector bundles.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Mathematics
Date of First Compliant Deposit:	12 July 2023
Last Modified:	06 Jan 2024 03:04
URI:	https://orca.cardiff.ac.uk/id/eprint/160969

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