Higher Grothendieck-Witt groups

这个项目的目的是加深我们对实拓扑K-理论的代数模拟：高阶Grothendieck-Witt群的理论的理解。目标是建立几个基本结果，特别强调消除对奇点和特征的限制，这些限制渗透在文献中。具体地说，我们将研究Karoubi和Williams的猜想，第一个猜想将无限正交的、辛的和一般的线性群的积分同调群联系起来，第二个猜想将K-理论的同伦不动点与Hermite K-理论联系起来。我们还将研究高阶Grothendieck-Witt群的同伦性质和设计性质，这在计算数域上整数环的Hermite K-群时是必不可少的。最后，我们将研究高阶Grothendieck-Witt群与由A^1-同伦理论定义的某些不变量的关系。从历史上看，上同调理论依附于几何对象，如地球表面(其拓扑性质在小变形下不改变)某些代数对象，如一组数字(本质上相当刚性)。对附加的代数对象的研究会产生有关原始几何对象的信息。上同调理论在拓扑学中的成功引导代数学家在代数的背景下定义上同调理论。这些代数上同调理论允许我们使用我们来自3空间的直觉和我们处理实数的经验来研究高维的系统多项式方程和在1+1可能是0的数字系统中(例如在密码学中使用)。本课题所研究的高阶Grothendieck-Witt群理论就是这样一种代数上同调理论。与它的同类理论--代数K-理论、Witt群和L-群相比，这个理论是相当不发达的。例如，与1+1=0的数制有关的几乎一无所知。这个项目旨在弥合高等Grothendieck-Witt群的1+理论和这些伴随理论之间的知识差距。

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 smalldeformations) 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 which1+1 = 0. This project aims to close the gap in knowledge between the 1+theory ofhigher Grothendieck-Witt groups and these companion theories.