Structures on Forms of K-Theory

Abstract

In the early 1970s, Morava studied forms of topological K-theory and observed that they have interesting number theoretic connections. Until very recently, forms of K-theory have not been studied in greater depth and integrated into the modern theory of topological modular forms. In this dissertation, some expected structured ring spectra and locality results are established on forms of K-theory. Forms of algebraic structures are usually classified by Galois cohomology. Based on the structured ring spectra and locality results established, a criterion is given for distinguishing homotopy equivalence classes of forms of K-theory via a computation in the second homotopy group of the spectrum.