Computation of algebraic K-theory of group rings R[G] is a problem of importance in both algebra and topology. In topology, the K-groups house many invariants associated to spaces with fundamental group G. One modern approach is through the study of so-called assembly maps. They relate the computable homology of the group G to it K-theory. The isomorphism conjecture asserts that the assembly maps are isomorphisms in all dimensions when the group has no torsion elements. The investigator will verify this for many groups that appear in geometry and geometric group theory. The method is to refine current techniques whenever they show such maps are injective and to use this refinement to prove surjectivity. In another direction, the investigator will develop techniques to handle more groups with combinatorial rather geometric prescriptions. One of the primary problems in topology is the classification of topological manifolds. An n-dimensional manifold is a space that locally looks like n-dimensional Euclidean space. Examples are spheres, tori, and other spaces often associated to such useful mathematical objects as solutions to differential equations. The most feasible approach is to classify manifolds within large subclasses known as homotopy classes, classification of such larger classes being the major goal of algebraic topology. Research of the investigator is directed toward resolution of the Borel rigidity conjecture, the simplest solution to this classification problem. The conjecture asserts that a certain natural type of manifold (technically, a compact manifold with a contractible cover) is unique within its homotopy class. If true, this rigidity, together with the effective ways to compare the homotopy classes in most relevant geometric situations, will result in a practical classification of these important manifolds. ***
