Hermitian K-theory of the Gaussian 2-integers

Abstract

In this thesis we the calculate the 2-completed homotopy type of the hermitian K-theory of the ring of Gaussian 2-integers. Motivation and background from number theory, algebra and algebraic geometry are also provided. We give introductions to classical K-theory and hermitian K-theory as well as the Quillen plus-construction. The hermitian K-groups of the Gaussian 2-integers are computed in degrees 0, 1 and 2 using these tools. For the homotopy type we use an étale version of hermitian K-theory, in the spirit of Dwyer-Friedlander, and construct a decomposition of the hermitian K-theory space of the Gaussian 2-integers into a product of spaces whose homotopy types are known. By work of Berrick, Karoubi and Østvær the étale hermitian K-theory of the Gaussian 2-integers is 2-adically equivalent to its ordinary hermitian K-theory.