eBook ISBN:  9781470404130 
Product Code:  MEMO/172/812.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 
eBook ISBN:  9781470404130 
Product Code:  MEMO/172/812.E 
List Price:  $79.00 
MAA Member Price:  $71.10 
AMS Member Price:  $47.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 172; 2004; 218 ppMSC: Primary 57; Secondary 20; 30
This text investigates a natural question arising in the topological theory of \(3\)manifolds, and applies the results to give new information about the deformation theory of hyperbolic \(3\)manifolds. It is well known that some compact \(3\)manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. We investigate when the subgroup \(\mathcal{R}(M)\) of outer automorphisms of \(\pi_1(M)\) which are induced by homeomorphisms of a compact \(3\)manifold \(M\) has finite index in the group \(\operatorname{Out}(\pi_1(M))\) of all outer automorphisms. This question is completely resolved for Haken \(3\)manifolds. It is also resolved for many classes of reducible \(3\)manifolds and \(3\)manifolds with boundary patterns, including all pared \(3\)manifolds.
The components of the interior \(\operatorname{GF}(\pi_1(M))\) of the space \(\operatorname{AH}(\pi_1(M))\) of all (marked) hyperbolic \(3\)manifolds homotopy equivalent to \(M\) are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to \(M\), so one may apply the topological results above to study the topology of this deformation space. We show that \(\operatorname{GF}(\pi_1(M))\) has finitely many components if and only if either \(M\) has incompressible boundary, but no “double trouble,” or \(M\) has compressible boundary and is “small.” (A hyperbolizable \(3\)manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli.) More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed.
Some expository sections detail Johannson's formulation of the JacoShalenJohannson characteristic submanifold theory, the topology of pared \(3\)manifolds, and the deformation theory of hyperbolic \(3\)manifolds. An epilogue discusses related open problems and recent progress in the deformation theory of hyperbolic \(3\)manifolds.
ReadershipGraduate students and research mathematicians interested in geometry and topology.

Table of Contents

Chapters

1. Introduction

2. Johannson’s characteristic submanifold theory

3. Relative compression bodies and cores

4. Homotopy types

5. Pared 3manifolds

6. Small 3manifolds

7. Geometrically finite hyperbolic 3manifolds

8. Statements of main theorems

9. The case when there is a compressible free side

10. The case when the boundary pattern is useful

11. Dehn flips

12. Finite index realization for reducible 3manifolds

13. Epilogue


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This text investigates a natural question arising in the topological theory of \(3\)manifolds, and applies the results to give new information about the deformation theory of hyperbolic \(3\)manifolds. It is well known that some compact \(3\)manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. We investigate when the subgroup \(\mathcal{R}(M)\) of outer automorphisms of \(\pi_1(M)\) which are induced by homeomorphisms of a compact \(3\)manifold \(M\) has finite index in the group \(\operatorname{Out}(\pi_1(M))\) of all outer automorphisms. This question is completely resolved for Haken \(3\)manifolds. It is also resolved for many classes of reducible \(3\)manifolds and \(3\)manifolds with boundary patterns, including all pared \(3\)manifolds.
The components of the interior \(\operatorname{GF}(\pi_1(M))\) of the space \(\operatorname{AH}(\pi_1(M))\) of all (marked) hyperbolic \(3\)manifolds homotopy equivalent to \(M\) are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to \(M\), so one may apply the topological results above to study the topology of this deformation space. We show that \(\operatorname{GF}(\pi_1(M))\) has finitely many components if and only if either \(M\) has incompressible boundary, but no “double trouble,” or \(M\) has compressible boundary and is “small.” (A hyperbolizable \(3\)manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli.) More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed.
Some expository sections detail Johannson's formulation of the JacoShalenJohannson characteristic submanifold theory, the topology of pared \(3\)manifolds, and the deformation theory of hyperbolic \(3\)manifolds. An epilogue discusses related open problems and recent progress in the deformation theory of hyperbolic \(3\)manifolds.
Graduate students and research mathematicians interested in geometry and topology.

Chapters

1. Introduction

2. Johannson’s characteristic submanifold theory

3. Relative compression bodies and cores

4. Homotopy types

5. Pared 3manifolds

6. Small 3manifolds

7. Geometrically finite hyperbolic 3manifolds

8. Statements of main theorems

9. The case when there is a compressible free side

10. The case when the boundary pattern is useful

11. Dehn flips

12. Finite index realization for reducible 3manifolds

13. Epilogue