Classical aspherical manifolds

F. Thomas Farrell and L. Edwin Jones0821807269

Aspherical manifolds–those whose universal covers are contractible–arise classically in many areas of mathematics. They occur in Lie group theory as certain double coset spaces and in synthetic geometry as the space forms preserving the geometry. This volume contains lectures delivered by the first author at an NSF-CBMS Regional Conference on K-Theory and Dynamics, held in Gainesville, Florida in January, 1989. The lectures were primarily concerned with the problem of topologically characterizing classical aspherical manifolds. This problem has for the most part been solved, but the 3- and 4-dimensional cases remain the most important open questions; Poincare’s conjecture is closely related to the 3-dimensional problem. One of the main results is that a closed aspherical manifold (of dimension $neq$ 3 or 4) is a hyperbolic space if and only if its fundamental group is isomorphic to a discrete, cocompact subgroup of the Lie group $O(n,1;{mathbb R})$. One of the book’s themes is how the dynamics of the geodesic flow can be combined with topological control theory to study properly discontinuous group actions on $R^n$. Some of the more technical topics of the lectures have been deleted, and some additional results obtained since the conference are discussed in an epilogue. The book requires some familiarity with the material contained in a basic, graduate-level course in algebraic and differential topology, as well as some elementary differential geometry.