K-Theory and Geometric Topology

Abstract

This is a survey article which explains the connection between K-theory in its various forms and its topological applications. The author chooses not to give a historical account, but rather a presentation which proceeds nicely in its logic, starting with K0 and obstructions to finiteness, through flat bundles, and to Whitehead and Reidemeister torsion. The survey continues with a discussion of controlled K-theory and negative K-groups and presents a brief introduction of Waldhausen's A-theory (which is a form of K-theory). The geometric application discussed here is pseudo-isotopies of manifolds. The survey ends with a rather unusual discussion of the connection between K-theory and symbolic dynamics. This survey is a pleasant read that can be recommended to graduate students. REVISED (August, 2006)