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
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