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

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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. ***