CBMS Regional Conference Series in Mathematics 1990; 54 pp; softcover Number: 75 ISBN10: 0821807269 ISBN13: 9780821807262 List Price: US$32 Member Price: US$25.60 All Individuals: US$25.60 Order Code: CBMS/75
 Aspherical manifoldsthose whose universal covers are contractiblearise 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 NSFCBMS Regional Conference on KTheory 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 4dimensional cases remain the most important open questions; Poincaré's conjecture is closely related to the 3dimensional 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, graduatelevel course in algebraic and differential topology, as well as some elementary differential geometry. Readership Table of Contents  The structure of manifolds from a historical perspective
 Flat Riemannian manifolds and infrasolvmanifolds
 The algebraic \(K\)theory of hyperbolic manifolds
 Locally symmetric spaces of noncompact type
 Existence of hyperbolic structures
 Epilogue
