Mathematical Sciences: Inverse Limit Problems in Algebraic K-Theory

本课题将研究代数K-理论中的两个重要结构。第一个是群环的K理论和L理论的集合图。该方法将是同伦理论，使用Pedersen和Webel的有界K-理论。研究者希望完成他对李群的余紧的、离散的、无挠的子群的满射性问题的研究，并推广这种方法以包括更一般的具有有限分类空间的群。第二个研究对象将是域F的代数K-理论的下降谱序列。他将研究在绝对伽罗华群是拓扑循环的情况下，F的代数闭包的K-理论的同伦有限构造。Colimit结构中的组成部分将是域F的K理论的副本。这些部分的细节各不相同，但每个部分都涉及将几何信息减少到用于计算的主题或完善用于计算的代数机器。涉及的几何信息的性质是困难的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有这些性质的系统研究，例如，如何区分两个几何对象在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些都只有在归结为计算问题时才被真正理解和掌握，而这方面的两个主要工具是同伦理论和K理论。

This project will study two important constructions in algebraic K-theory. The first is the assembly map for the K-theory and L-theory of group rings. The method will be homotopy- theoretic, using the bounded K-theory of Pedersen and Weibel. The investigator hopes to complete his study of the surjectivity question for cocompact, discrete, torsion-free subgroups of Lie groups, and to generalize the approach to include more general groups with finite classifying space. The second object of study will be the descent spectral sequence for the algebraic K-theory of a field F. He will study a homotopy colimit construction for the K-theory of the algebraic closure of F, in the case where the absolute Galois group is topologically cyclic. The component pieces in the colimit construction will be copies of the K-theory of the field F. The details of these parts vary, but each is concerned with reducing geometric information to a subject for calculation or to perfecting the algebraic machinery used for the calculations. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation, and two of the principal tools for this are homotopy theory and K-theory.