Classical and A1-homotopy theory of linear algebraic groups

本课题研究代数与拓扑学之间的接口。研究了代数结构对称群的同伦理论，即连续变形后不改变的性质。该提案分为两部分。第一种情况是，当代数对象由k域上的向量空间组成时，没有额外的结构。在这种情况下，该结构将由nxn个可逆矩阵组成的一般线性群GLn分组。研究了GLn的a1 -同伦理论。a1 -同伦是定义代数定义对象的同伦理论的一种强有力的方法。在这个最早建立于20世纪90年代末的理论中，人们考虑那些可能由多项式函数定义的变形。在经典同伦理论中，关于空间X的许多信息都被编码在它的同伦群：pi_n(X)中，这些同伦群记录了从球到X的连续函数的不同同伦类。在a1同伦中，可以类似地定义同伦群，但现在的同伦意义是a1同伦。普通同伦群在大多数情况下很难计算，而a1 -同伦群更是难以确定。在k的代数k理论的幌子下，GLn的a1 -同伦群和相关空间编码了关于底层域k的微妙而神秘的信息，本提案将计算这些同伦群以提取和理解这些信息。我们将深入了解代数几何对象X上的向量束理论与X的代数k理论之间的关系。我们还将更多地了解球本身的同伦群，因为群GLn是a1 -同伦球a ^n-0的对称群。提案的第二部分考察了当向量空间A有一个额外的结构（如乘法）时会发生什么。那么A就是一个代数，一个数学中的普遍结构。对称G受到A的乘法的限制，比没有乘法的情况更难理解。有一些特殊的几何空间与（G，A）的数据相关：空间参数化A中元素的r-元组，足以生成A的整个代数结构。这些空间迄今为止很少被研究，但由于它们是代数定义的，我们可以使用代数技术来检查它们的普通同伦理论，促进许多显式计算。通过这种方式，我们将阐明对称群G和与a相关的代数结构。该项目将使用同伦理论来加深我们对几种不同类型的广泛使用的代数结构的基础知识：代数、对合代数、代数对象上的向量束和场（通过k理论）。它还将告诉我们更多关于球体之间映射的拓扑，这是最基本的拓扑对象，但许多问题仍未得到解答。

This project studies the interface between algebra and topology. We study the homotopy theory, i.e., the properties that do not change even after continuous deformations, of the symmetry groups of algebraic structures. The proposal is in two parts. The first is when the algebraic objects consist of vector spaces over a field k with no additional structure. In this case, the structure groups the general linear groups GLn, which are comprised of nxn invertible matrices. We study the A1-homotopy theory of GLn. A1-homotopy is a powerful way to define a homotopy theory of algebraically-defined objects. In this theory, first established in the late 1990s, one considers those deformations that may be defined by polynomial functions. In classical homotopy theory, much information about a space X is encoded in its homotopy groups: pi_n(X), which record the different homotopy-classes of continuous functions from spheres to X. In A1-homotopy, one may analogously define homotopy groups, but now the sense of homotopy is the A1-homotopy. The ordinary homotopy groups are difficult to calculate in most cases, and the A1-homotopy groups are even more difficult to determine. The A1-homotopy groups of GLn and related spaces encode subtle and mysterious information about the underlying field k, in the guise of the algebraic K-theory of k, and this proposal will calculate these homotopy groups in order to extract and make sense of that information. We will gain insight into the way in which the theory of vector bundles over an algebraic-geometric object X relates to the algebraic K-theory of X. We will also learn more about the homotopy groups of the spheres themselves, since the group GLn is a symmetry group of the A1-homotopy sphere A^n-0. The second part of the proposal examines what happens when the vector space A has an additional structure, such as multiplication. A is then an algebra, a prevalent structure in mathematics. The symmetries G are restricted by the multiplication of A and are harder to understand than in the case where the multiplication was absent. There are particular geometric spaces associated to the data of (G,A): spaces parametrizing r-tuples of elements in A that are sufficient to generate the entire algebraic structure of A. These spaces are little-studied to date, but because they are algebraically defined, we may use algebraic techniques to examine their ordinary homotopy theory, facilitating a number of explicit calculations. In this way, we will cast light on the symmetry group G and on algebraic structures related to A. The project will use homotopy theory to deepen our fundamental knowledge about several different kinds of widely-used algebraic structures: algebras, algebras with involution, vector bundles on algebraic objects, and fields (through the K-theory). It will also tell us more about the topology of maps between spheres, which are the most fundamental topological objects but about which many questions remain unanswered.