Plus Constructions, Assembly Maps and Group Actions on Manifolds

Abstract

The Farrell-Jones conjecture says that the algebraic K-groups of a group ring can be computed by the equivariant homology groups of a classifying space via an assembly map. Therefore, to address this conjecture it is important to understand the definition of K-theory, equivariant homology theory and group actions (for providing models of classifying spaces). The thesis consists of three parts. The first part aims to understand the definitions of algebraic K-theory. We introduce a construction adding low-dimensional cells (handles) to a CW complex (manifold) that satisfi es certain low-dimensional conditions. It preserves high-dimensional homology with appropriate coeffi cients. This includes as special cases Quillen's plus construction, Bous eld's integral homology localization, Varadarajan's existence of Moore spaces M(G; 1), Bous eld and Kan's partial k-completion of spaces, the existences of high dimensional knot groups and homology spheres proved by Kervaire. We also use the construction to get some examples for the zero-in-the-spectrum conjecture, which give generalizations of the examples found by Farber-Weinberg and Higson-Roe-Schick. The second part investigates the equivariant homology theory. We give a computation of equivariant homology theories over categories. This generalizes both Arlettaz's result for generalized homology theory and L ueck's rational computation of Chern characters for equivariant K-theory. Some applications to algebraic K-theory are obtained as well. We prove that for a fi xed group, there is an injection of the homology groups of the group into the algebraic K-groups of the group ring, after tensoring with some subring of rationals. The third part studies matrix group actions on CAT(0) spaces and manifolds. It is shown that matrix groups can only act trivially on low-dimensional spheres and that matrix group actions on low-dimensional CAT(0) spaces always have a global fixed point. These results give generalizations of results obtained by Bridson-Vogtmann and Pawani concerning special linear group actions on spheres and of results obtained by Farb concerning Chevalley group actions on CAT(0) spaces. As applications to low-dimensional representations, we show that there are no non-trivial group homomorphisms from matrix groups to low-sized matrix groups for some rings.