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Classical Aspherical Manifolds

A co-publication of the AMS and CBMS
Available Formats:
Electronic ISBN: 978-1-4704-2435-0
Product Code: CBMS/75.E
List Price: $32.00 Individual Price:$25.60
Click above image for expanded view
Classical Aspherical Manifolds
A co-publication of the AMS and CBMS
Available Formats:
 Electronic ISBN: 978-1-4704-2435-0 Product Code: CBMS/75.E
 List Price: $32.00 Individual Price:$25.60
• Book Details

CBMS Regional Conference Series in Mathematics
Volume: 751990; 54 pp
MSC: Primary 18; 22; 53; 57; Secondary 19;

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.

Readership

• Table of Contents

• Chapters
• The Structure of Manifolds from a Historical Perspective
• Fiat Riemannian Manifolds and Infrasolvmanifolds
• The Algebraic $K$-theory of Hyperbolic Manifolds
• Locally Symmetric Spaces of Noncompact Type
• Existence of Hyperbolic Structures
• Epilogue
• Requests

Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
Volume: 751990; 54 pp
MSC: Primary 18; 22; 53; 57; Secondary 19;

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.

Readership

• Chapters
• The Structure of Manifolds from a Historical Perspective
• Fiat Riemannian Manifolds and Infrasolvmanifolds
• The Algebraic $K$-theory of Hyperbolic Manifolds
• Locally Symmetric Spaces of Noncompact Type
• Existence of Hyperbolic Structures
• Epilogue
Review Copy – for reviewers who would like to review an AMS book
Accessibility – to request an alternate format of an AMS title
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