Mathematical Sciences: Manifolds and Algebraic K-Theory

本研究项目分为三部分。第一部分涉及到稳定范围外流形的微分同态的同伦类型的一些思想，这些思想与代数k理论有关。第二部分是用循环子群来理解群环的k理论的程序。第三是关于对K(Z)和A（S1）的切眼迹的应用。这三个部分的细节各不相同，但都涉及将几何信息简化为一个计算主题，或者完善用于此目的的主要代数工具之一。所涉及的几何信息的性质是难点的关键。虽然关于长度、面积、角度、体积等问题实际上迫切需要简化为计算，但这与几何对象的拓扑特性大不相同。这些属性包括连通性（整体）、打结性、无孔等等。所有对这些性质的系统研究，例如，如何判断两个几何物体在其中一个性质上是否真的不同，或者只是表面上的不同，或者如何对可能出现的各种差异进行分类，所有这些只有在它们被简化为计算问题时才真正被理解和掌握。代数k理论已经发展成为这一目的的主要工具，代数和拓扑之间的相互作用仍然是一个迷人的主题。

This research project has three parts. The first involves some ideas about the homotopy type of diffeomorphisms of manifolds beyond the stable range, in which these are related to algebraic K-theory. The second is a program for understanding the K-theory of group rings in terms of cyclic subgroups. The third is concerned with applications of the cyclotomic trace to K(Z) and to A(S1). The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. 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. Algebraic K-theory has been developed into a major tool for this purpose, and the interplay between the algebra and the topology involved remains a fascinating subject.