# Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups

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*Richard D. Canary; Darryl McCullough*

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
Jaco-Shalen-Johannson 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.

#### Table of Contents

# Table of Contents

## Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups

- Contents v6 free
- Preface ix10 free
- Chapter 1. Introduction 114 free
- Chapter 2. Johannson's Characteristic Submanifold Theory 1528
- 2.1. Fibered 3-manifolds 1629
- 2.2. Boundary patterns 2033
- 2.3. Admissible maps and mapping class groups 2336
- 2.4. Essential maps and useful boundary patterns 2841
- 2.5. The classical theorems 3548
- 2.6. Exceptional fibered 3-manifolds 3851
- 2.7. Vertical and horizontal surfaces and maps 4053
- 2.8. Fiber-preserving maps 4154
- 2.9. The characteristic submanifold 4861
- 2.10. Examples of characteristic submanifolds 5164
- 2.11. The Classification Theorem 5770
- 2.12. Miscellaneous topological results 6073

- Chapter 3. Relative Compression Bodies and Cores 6578
- Chapter 4. Homotopy Types 7790
- Chapter 5. Pared 3-Manifolds 87100
- Chapter 6. Small 3-Manifolds 97110
- Chapter 7. Geometrically Finite Hyperbolic 3-Manifolds 105118
- Chapter 8. Statements of Main Theorems 117130
- Chapter 9. The Case When There Is a Compressible Free Side 121134
- Chapter 10. The Case When the Boundary Pattern Is Useful 139152
- Chapter 11. Dehn Flips 175188
- Chapter 12. Finite Index Realization For Reducible 3-Manifolds 179192
- Chapter 13. Epilogue 195208
- Bibliography 207220
- Index 213226 free