Surgery L-Groups, Algebraic K-Groups and Rigidity of Classical Aspherical Manifolds

Borel猜想指出任何非球面闭流形都是由它的基群在拓扑上决定的。本研究项目将提供一种新的几何工具，使研究者能够对任何非球闭流形的基本群与非正弯曲黎曼流形的基本群同构的非球闭流形进行验证。几何工具由几何学家Cheeger-Fukaya-Gromov等人提出的几何坍缩理论的推广组成。在这个项目中，L. E. Jones（与F. T. Farrell合作）将发展叶状黎曼流形的坍缩理论。“二维流形”是一种“空间”，对于接近空间中任何一点的观察者来说，它看起来完全像欧几里得平面，但当作为一个整体观察时，它可能根本不是欧几里得空间。例如，如果我们实际上在地球表面上，地球表面看起来像一个平面，但从太空观察时，它被视为一个球体。19世纪的数学家已经很好地理解了二维流形。还有三维流形，四维流形，等等，这些流形不太为人所知；事实上，数学家对高维流形的探索激发了20世纪的大部分拓扑学研究。研究流形本质的一种方法是关注一个相关的代数对象（这更容易理解），称为它的“基本群”。一个著名的数学猜想指出，对于一类特殊的高维流形（称为“非球面流形”），基本群应该充当流形的遗传密码，揭示其所有内部拓扑秘密。这个项目的目的是验证这个猜想。＊＊＊

9704765 Jones The Borel Conjecture states that any aspherical closed manifold should be topologically determined by its fundamental group. This research project will provide a new geometric tool that will enable the investigator to verify this conjecture for any aspherical closed manifold whose fundamental group is isomorphic to the fundamental group of a non-positively curved Riemannian manifold. The geometric tool consists of a generalization of the geometric collapsing theory developed by the geometers Cheeger-Fukaya-Gromov and others. In this project L. E. Jones (in collaboration with F. T. Farrell) will develop a collapsing theory for foliated Riemannian manifolds. A "two-dimensional manifold" is a "space" that looks exactly like the Euclidean plane to an observer near any point in the space, but which when observed as a whole may not be Euclidean space at all. For example the earth's surface looks like a flat plane if we are actually on its surface, but when observed from space, it is seen to be a sphere. Two-dimensional manifolds were already well understood by mathematicians of the 19th century. There are also 3-dimensional manifolds, 4-dimensional manifolds, etc., manifolds that are less well understood; in fact, the mathematician's quest to understand higher dimensional manifolds has motivated much of the topological research of the 20th century. One way to investigate the nature of a manifold is to focus on an associated algebraic object (that is much easier to understand) called its "fundamental group." A famous mathematical conjecture states that for a special class of higher dimensional manifolds (called "aspherical manifolds"), the fundamental group should act like a genetic code for the manifold, revealing all of its inner topological secrets. The aim of this project is to verify this conjecture. ***