Preface v Chapter 1. Elliptic 3-manifolds and the Smale Conjecture 1.1. Elliptic 3-manifolds and their isometries 1.2. The Smale Conjecture 1.3. Isometries of nonelliptic 3-manifolds 1.4. Perelman's methods Chapter 2. Diffeomorphisms and embeddings of manifolds 2.1. The C ∞ -topology 2.2. Metrics which are products near the boundary 2.3. Manifolds with boundary 2.4. Spaces of embeddings 2.5. Bundles and fiber-preserving diffeomorphisms 2.6. Aligned vector fields and the aligned exponential Chapter 3. The method of Cerf and Palais 3.1. The Palais-Cerf Restriction Theorem 3.2. The space of images 3.3. Projection of fiber-preserving diffeomorphisms 3.4. Restriction of fiber-preserving diffeomorphisms 3.5. Restriction theorems for orbifolds 3.6. Singular fiberings 3.7. Spaces of fibered structures 3.8. Restricting to the boundary or the basepoint 3.9. The space of Seifert fiberings of a Haken 3-manifold 3.10. The Parameterized Extension Principle Chapter 4. Elliptic 3-manifolds containing one-sided Klein bottles 4.1. The manifolds M(m, n) 4.2. Outline of the proof 4.3. Isometries of elliptic 3-manifolds 4.4. The Hopf fibering of M(m, n) and special Klein bottles 4.5. Homotopy type of the space of diffeomorphisms 4.6. Generic position configurations iii iv CONTENTS 4.7. Generic position families 4.8. Parameterization Chapter 5. Lens spaces 5.1. Outline of the proof 5.2. Reductions 5.3. Annuli in solid tori 5.4. Heegaard tori in very good position 5.5. Sweepouts, and levels in very good position 5.6. The Rubinstein-Scharlemann graphic 5.7. Graphics having no unlabeled region 5.8. Graphics for parameterized families 5.9. Finding good regions 5.10. From good to very good 5.11. Setting up the last step 5.12. Deforming to fiber-preserving families 5.13. Parameters in D d Bibliography Index PrefaceThis work is ultimately directed at understanding the diffeomorphism groups of elliptic 3-manifolds-those closed 3-manifolds that admit a Riemannian metric of constant positive curvature. The main results concern the Smale Conjecture. The original Smale Conjecture, proven by A. Hatcher [24], asserts that if M is the 3-sphere with the standard constant curvature metric, the inclusion Isom(M) → Diff(M) from the isometry group to the diffeomorphism group is a homotopy equivalence. The Generalized Smale Conjecture (henceforth just called the Smale Conjecture) asserts this whenever M is an elliptic 3-manifold.Here are our main results:1. The Smale Conjecture holds for elliptic 3-manifolds containing geometrically incompressible Klein bottles (Theorem 1.2.2). These include all quaternionic and prism manifolds. 2. The Smale Conjecture holds for all lens spaces L(m, q) with m ≥ 3 (Theorem 1.2.3).Many of the cases in Theorem 1.2.2 were proven a number of years ago by N. Ivanov [32,34,35,36] (see Section 1.2). Some of our other results concern the groups of diffeomorphisms Diff(Σ) and fiber-preserving diffeomorphisms Diff f (Σ) of a Seifertfibered Haken 3-manifold Σ, and the coset space Diff(Σ)/ Diff f (Σ), which is called the space of Seif...

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For a Haken 3-manifold M with incompressible boundary, we prove that the mapping class group M acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in M is geometrically finite. Also, a simplified proof of the fact that torsionfree subgroups of finite index in M exist is given. All results are given for mapping class groups that preserve a boundary pattern in the sense of K. Johannson. As an application, we show that if F is a nonempty compact 2-manifold in ∂M such that ∂M − F is incompressible, then the classifying space BDif f (M rel F ) of the diffeomorphism group of M relative to F has the homotopy type of a finite aspherical complex.

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