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

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Homotopy Equivalences of 3-Manifolds and Deformation Theory of Kleinian Groups Richard D. Canary Darryl McCullough Author address: Department of Mathematics University of Michigan Ann Arbor, MI 48109 E-mail address: [email protected] URL: www.math.lsa.umich.edu/ canary/ e Department of Mathematics University of Oklahoma Norman, OK 73019 E-mail address: [email protected] URL: www.math.ou.edu/ dmccullough/ e Contents Preface ix Chapter 1. Introduction 1 1.1. Motivation 1 1.2. The main theorems for Haken 3-manifolds 3 1.3. The main theorems for reducible 3-manifolds 8 1.4. Examples 9 Chapter 2. Johannson's Characteristic Submanifold Theory 15 2.1. Fibered 3-manifolds 16 2.2. Boundary patterns 20 2.3. Admissible maps and mapping class groups 23 2.4. Essential maps and useful boundary patterns 28 2.5. The classical theorems 35 2.6. Exceptional fibered 3-manifolds 38 2.7. Vertical and horizontal surfaces and maps 39 2.8. Fiber-preserving maps 41 2.9. The characteristic submanifold 48 2.10. Examples of characteristic submanifolds 51 2.11. The Classification Theorem 57 2.12. Miscellaneous topological results 59 Chapter 3. Relative Compression Bodies and Cores 65 3.1. Relative compression bodies 66 3.2. Minimally imbedded relative compression bodies 69 3.3. The maximal incompressible core 71 3.4. Normally imbedded relative compression bodies 73 3.5. The normal core and the useful core 74 Chapter 4. Homotopy Types 77 4.1. Homotopy equivalences preserve usefulness 77 4.2. Finiteness of homotopy types 83 Chapter 5. Pared 3-Manifolds 87 5.1. Definitions and basic properties 87 5.2. The topology of pared manifolds 89 5.3. The characteristic submanifold of a pared manifold 92 Chapter 6. Small 3-Manifolds 97 6.1. Small manifolds and small pared manifolds 97 6.2. Small pared homotopy types 101 v vi CONTENTS Chapter 7. Geometrically Finite Hyperbolic 3-Manifolds 105 7.1. Basic definitions 105 7.2. Quasiconformal deformation theory: a review 107 7.3. The Parameterization Theorem 114 Chapter 8. Statements of Main Theorems 117 8.1. Statements of Main Topological Theorems 117 8.2. Statements of Main Hyperbolic Theorem and Corollary 118 8.3. Derivation of hyperbolic results 119 Chapter 9. The Case When There Is a Compressible Free Side 121 9.1. Algebraic lemmas 122 9.2. The finite-index cases 124 9.3. The infinite-index cases 132 Chapter 10. The Case When the Boundary Pattern Is Useful 139 10.1. The homomorphism Ψ 141 10.2. Realizing homotopy equivalences of I-bundles 153 10.3. Realizing homotopy equivalences of Seifert-fibered manifolds 161 10.4. Proof of Main Topological Theorem 2 171 Chapter 11. Dehn Flips 175 Chapter 12. Finite Index Realization For Reducible 3-Manifolds 179 12.1. Homeomorphisms of connected sums 180 12.2. Reducible 3-manifolds with compressible boundary 190 12.3. Reducible 3-manifolds with incompressible boundary 191 Chapter 13. Epilogue 195 13.1. More topology 195 13.2. More geometry 197 Bibliography 207 Index 213 Abstract 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 homeo- morphisms. We investigate when the subgroup (M) of outer automorphisms of R π1(M) which are induced by homeomorphisms of a compact 3-manifold M has fi- nite index in the group Out(π1(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 GF(π1(M)) of the space AH(π1(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 GF(π1(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 re- lated open problems and recent progress in the deformation theory of hyperbolic 3-manifolds. Received by the editor February 6, 2001. 2000 Mathematics Subject Classification. Primary 57M99; Secondary 20F34, 30F40, 57M50. Key words and phrases. 3-manifold, hyperbolic, pared, reducible, convex-cocompact, geo- metrically finite, boundary pattern, Haken, reducible, characteristic submanifold, compression body, homotopy equivalence, homeomorphism, diffeomorphism, automorphism, double trouble. Supported in part by the National Science Foundation and the Sloan Foundation. Supported in part by the National Science Foundation. vii Preface This work addresses a question about homotopy equivalences and homeomor- phisms of 3-manifolds, and gives an application to the topology of deformation spaces of hyperbolic 3-manifolds. Although the topological question is quite natu- ral, it did not receive much attention until motivated by the geometric application. It is simply this: in the group of homotopy classes of self-homotopy-equivalences of a 3-manifold, when does the subgroup consisting of the classes that contain a homeomorphism have finite index? Most of our results will apply only to Haken 3-manifolds, although in chap- ter 12 we develop rather general versions of our theorems for reducible 3-manifolds. For closed Haken manifolds, Waldhausen proved that all homotopy classes con- tain homeomorphisms, so we need only consider Haken manifolds with nonempty boundary. This case breaks into two very different subcases, the manifolds with incompressible boundary and those with compressible boundary. We resolve the question completely, giving exact conditions for the subgroup of classes realizable by homeomorphisms to have finite index. In the case when the boundary is incompressible, our principal means to ex- amine homotopy equivalences and homeomorphisms of Haken 3-manifolds is the characteristic submanifold theory due independently to Jaco-Shalen and Johann- son. For our geometric application, we need to consider a more general version of the question, in which we work with maps that preserve certain submanifolds of the boundary. The formulation of the characteristic submanifold given by Johannson, with its elegant theory of boundary patterns, provides exactly the kind of control that we need. For this reason, almost all of our work is carried out in the context of manifolds with boundary patterns. A complete resolution of the realization ques- tion for Haken manifolds with boundary patterns seems out of reach, and our main results are restricted to the case when the submanifolds forming the boundary pat- tern are disjoint. But these do include as a very special case the boundary patterns which arise in the study of the deformation theory of hyperbolic 3-manifolds. The statements of the main results require a number of preliminary concepts. Consequently, they cannot conveniently be given until rather late in the develop- ment of our exposition, indeed not until chapter 8. To motivate so much preliminary work, we state and discuss restricted versions of the main results in chapter 1. There we lay out the general program and provide several examples illustrating some of the phenomena that can occur. Chapter 2 contains an exposition of Johannson's version of the characteristic submanifold theory. We review the concepts which underlie his main results, give a variety of examples of boundary patterns and characteristic submanifolds, and develop some technical results which we need for our later work. All of Johannson's theory takes place under the assumption that the 3-manifold satisfies a certain ix x PREFACE incompressibility condition on its boundary (namely, that the boundary pattern has the property of being useful). To work with manifolds that do not satisfy this condition (that is, to allow some boundary patterns that are not useful, in the technical sense), we use a second kind of characteristic structure, the characteristic compression body invented by Bonahon. For manifolds with boundary patterns, we need a relative version of his theory, which we develop in chapter 3. (Both the characteristic compression body and our relative version are generalizations of the Abikoff-Maskit decomposition which was developed earlier in the context of Kleinian groups.) With these topological concepts in place, we can prove in chapter 4 some results about homotopy equivalences and homotopy types of 3- manifolds with boundary patterns, which generalize theorems of Johannson and Swarup. Geometry enters, at least implicitly, in chapter 5, where we introduce the boundary patterns naturally associated to manifolds with hyperbolic structures. These are called pared structures. We review the definitions and some known results, including the strong restrictions on the structure of the characteristic sub- manifold associated to a pared structure. In chapter 6, we define small 3-manifolds. These arise as exceptional cases for our main topological results, in the case where the manifold has compressible boundary. We determine which of the small mani- folds have pared structures, and prove a technical result which lists the homotopy types all of whose homeomorphism classes are either compression bodies or small manifolds. The actual geometric theory which we use draws on work of many mathemati- cians, including Ahlfors, Bers, Kra, Marden, Maskit, Sullivan, and Thurston.
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