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

9401058法雷尔 F.T. Farrell与L.E.琼斯（纽约州立大学石溪分校）将试图验证他们的代数计算的K-和L-理论的整群环的一个给定的群体的K-和L-理论的整群环的一类子群，该组。 这个类由给定群的所有虚循环子群组成。 他们设想应用这项工作的微分几何，特别是刚性的问题。 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，空间局部类似于欧几里得空间，除了距离的概念可能被扭曲，它们在物理理论的数学表述中自然而普遍地出现。 因此，正如从广义相对论中所熟悉的那样，空间可以是弯曲的;例如，它可以是球体或环面（环形表面）。 这种空间的一些基本数值不变量是它的基本群和它的截面曲率。 例如，环面的曲率从一点到另一点连续变化，在正数和负数之间变化，而半径为1的球体的曲率相同。 Farrell和Jones证明了两个具有相同基本群不变量的闭负弯曲黎曼流形是同胚的;即，在它们的点之间存在连续的一对一映射。 (The 3维和4维流形的情况仍然是开放的。 但他们给出了映射永远不会平滑的例子。 搜索将保证一个光滑的映射，这是非常重要的任何物理应用额外的数值不变量。 ***

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. ***