The aim of this project is to deepen our understanding of the algebraic analogue of real topological K-theory: the theory of higher Grothendieck-Witt groups. The goal is to establish several fundamental results with special emphasis on eliminating restrictions on singularities and characteristics which permeate the literature. Specifically, we will study conjectures of Karoubi and Williams - the first, relating the integral homology groups of infinite orthogonal, symplectic and general linear groups, and the second relating homotopy fixed points of K-theory to hermitian K-theory. We will also study homotopy and devissage properties of higher Grothendieck-Witt groups when "2 is not invertible" which are essential in the calculation of hermitian K-groups of rings of integers in number fields. Finally, we will study higher Grothendieck-Witt groups in relation with certain invariants defined via A^1-homotopy theory.
Historically, cohomology theories attach to a geometric object such as the surface of the earth (whose topological properties don't change under small deformations) certain algebraic objects such as a set of numbers (which are rather rigid in nature). The study of the attached algebraic objects yields information about the original geometric object. The success of cohomology theories in topology lead algebraists to define cohomology theories in an algebraic context. These algebraic cohomology theories allow us to use our intuition from 3 space and our experience with working with real numbers to study systems polynomial equations in higher dimensions and in number systems (used e.g. in cryptography) where 1+1 could be 0. The theory investigated in this project, the theory of higher Grothendieck-Witt groups, is one such algebraic cohomology theory. Compared to its companion theories - algebraic K-theory, Witt-groups and L-groups - this theory is rather underdeveloped. For instance, virtually nothing is known in relation with number systems in which 1+1 = 0. This project aims to close the gap in knowledge between the 1+theory of higher Grothendieck-Witt groups and these companion theories.
