Higher Grothendieck-Witt groups

这个项目的目的是加深我们对真实的拓扑K-理论的代数类似物的理解：高等Grothendieck-Witt群的理论。目标是建立几个基本结果，特别强调消除对奇异性和特征的限制，这些限制渗透到文献中。具体地说，我们将研究Karoubi和威廉姆斯的结构-第一，有关的积分同调群的无限正交，辛和一般线性群，第二有关同伦不动点的K-理论，埃尔米特K-理论。我们还将研究高阶Grothendieck-Witt群在“2不可逆”时的同伦性和设计性，这是数域上整数环的Hermitian K-群的计算中必不可少的。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。历史上，上同调理论与几何对象（如地球表面，其拓扑性质在小变形下不会改变），某些代数对象（如一组数，其性质相当严格）有关。对附加代数对象的研究产生关于原始几何对象的信息。上同调理论在拓扑学中的成功使得代数学家在代数学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的直觉从3空间和我们的经验与工作与真实的数字研究系统多项式方程在更高的维度和数字系统（用于例如在密码学），其中1+1可以是0。在这个项目中研究的理论，高阶Grothendieck-Witt群的理论，就是这样一个代数上同调理论。与它的同伴理论--代数K-理论、维特群和L-群--相比，这个理论还相当不发达。例如，实际上我们对1 +1 = 0的数制一无所知。本项目旨在缩小高等Grothendieck-Witt群的1+理论与这些同伴理论之间的知识差距差距。

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.