Mathematical Sciences: Some Problems on the Interface Between Geometry and Topology

9401058 Farrell F.T.Farrell和L.E.Jones(纽约州立大学斯坦布鲁克分校)将尝试根据给定群的一类子群的积分群环的K-和L-理论来验证他们对给定群的积分群环的代数K-和L-理论的猜想计算。这个类由给定群的所有虚拟循环子群组成。他们设想将这项工作应用于微分几何，特别是刚性问题。目标是通过数值不变量对几何对象进行分类。几何对象是黎曼流形，即局部类似于欧几里德空间的空间，除了距离的概念可能被扭曲外，它们自然地和普遍地出现在物理理论的数学公式中。因此，正如广义相对论所熟知的那样，空间可以是曲面的；例如，它可以是球体或环面(甜甜圈曲面)。这种空间的一些基本数值不变量是它的基本群和截面曲率。例如，环面的曲率从一个点到另一个点连续变化，在正负数之间变化，而半径为1的球面的曲率恒定为1。Farrell和Jones证明了两个具有相同基本群不变量的闭负曲黎曼流形是同胚的，即它们的点之间存在连续的一对一映射。(三维和四维流形的情况仍未解决。)但他们给出了映射永远不可能顺利的例子。将搜索保证平滑映射的其他数值不变量，这对于任何物理应用都是非常重要的。***

9401058 Farrell F.T. Farrell in collaboration with L.E. Jones (SUNY at StonyBrook) will attempt to verify their conjectured calculation of the algebraic K- and L-theories of the integral group ring of a given group in terms of the K- and L-theories of the integral group rings of a class of subgroups of that group. This class consists of all the virtually cyclic subgroups of the given group. They envision applications of this work to differential geometry, in particular, to questions of rigidity. The objective is to classify geometric objects by numerical invariants. The geometric objects are Riemannian manifolds, spaces which locally resemble Euclidean space except that the notion of distance may be warped, and they arise naturally and ubiquitously in the mathematical formulation of physical theories. Thus, as is familiar from the general theory of relativity, a space can be curved; e.g., it could be a sphere or a torus (doughnut surface). Some basic numerical invariants of such a space are its fundamental group and its sectional curvatures. For example, the curvature of the torus changes continuously from point to point, varying between positive and negative numbers, whereas the curvature of the sphere of radius 1 is identically 1. Farrell and Jones have shown that two closed negatively curved Riemannian manifolds with the same fundamental group invariant are homeomorphic; i.e., there is a continuous one-to-one mapping between their points. (The case of 3- and 4-dimensional manifolds is still open.) But they gave examples where the mapping can never be smooth. A search will be made for additional numerical invariants guaranteeing a smooth mapping, which is highly significant for any physical applications. ***