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...

ABSTRACT. An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.

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The symmetries of manifolds are a focal point of study in low-dimensional topology and yet, outside of some totally asymmetrical 3- and 4-manifolds, there are very few cases in which a complete classification has been attained. In this work we provide such a classification for symmetries of the orientable and nonorientable 3-dimensional handlebodies of genus one. Our classification includes a description, up to isomorphism, of all of the finite groups which can arise as symmetries on these manifolds, as well as an enumeration of the different ways in which they can arise. To be specific, we will classify the equivalence, weak equivalence and strong equivalence classes of (effective) finite group actions on the genus one handlebodies.

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