Date of Defense
8-4-2026 12:00 PM
Location
F3-238
Document Type
Thesis Defense
Degree Name
Master of Science in Mathematics
College
COS
Department
Mathematical Sciences
First Advisor
Prof. Ahmed Al Rawashdeh
Keywords
Category theory, Equivalence of categories, Vector bundles, Projective modules, K-theory.
Abstract
This thesis explores topological K-theory, a powerful framework for studying invariants of topological spaces via algebraic structures. We investigate the construction of K-groups, with particular emphasis on the categorical formulation. The exposition is structured pedagogically, with careful development of the main concepts supported by detailed proofs and examples. The categories of vector bundles and projective modules are thoroughly defined and investigated. A central result studied in this thesis is the Serre–Swan theorem, which provides a bridge between the algebraic context of projective modules and the geometric context of vector bundles, enabling the use of algebraic techniques to solve geometric problems, and vice versa. The thesis concludes with the study of a fundamental structural result in K-theory, the Bott Periodicity theorem. As an application of this theorem, we present an alternative proof of Brouwer’s Fixed Point theorem.
Included in
ON TOPOLOGICAL AND ALGEBRAIC K-THEORIES
F3-238
This thesis explores topological K-theory, a powerful framework for studying invariants of topological spaces via algebraic structures. We investigate the construction of K-groups, with particular emphasis on the categorical formulation. The exposition is structured pedagogically, with careful development of the main concepts supported by detailed proofs and examples. The categories of vector bundles and projective modules are thoroughly defined and investigated. A central result studied in this thesis is the Serre–Swan theorem, which provides a bridge between the algebraic context of projective modules and the geometric context of vector bundles, enabling the use of algebraic techniques to solve geometric problems, and vice versa. The thesis concludes with the study of a fundamental structural result in K-theory, the Bott Periodicity theorem. As an application of this theorem, we present an alternative proof of Brouwer’s Fixed Point theorem.