# Classical Aspherical Manifolds

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*Lowell E. Jones; F. Thomas Farrell*

A co-publication of the AMS and CBMS

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; Poincaré'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.

#### Table of Contents

# Table of Contents

## Classical Aspherical Manifolds

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction vii8 free
- 1. The Structure of Manifolds from a Historical Perspective 110 free
- 2. Flat Riemannian Manifolds and Infrasolvmanifolds 1221
- 3. The Algebraic K-theory of Hyperbolic Manifolds 1928
- 4. Locally Symmetrie Spaces of Noncompact Type 3039
- 5. Existence of Hyperbolic Structures 3645
- 6. Epilogue 4756
- References 5059
- Back Cover Back Cover164