Diffeomorphisms of Elliptic 3-Manifolds

Abstract

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...

Diffeomorphisms preserving Morse–Bott functions

“…for all closed hyperblic 3-manifolds (D. Gabai [41]), the unit 3-sphere S 3 with the standard metric (A. Hatcher [42]), and more generally for many classes of elliptic 3-manifolds, e.g. [43]. Chapter 1 of the latter book [43] also contains a comprehensive historical description and the current state of Smale conjecture.…”

“…[43]. Chapter 1 of the latter book [43] also contains a comprehensive historical description and the current state of Smale conjecture.…”

Banchoff’s sphere and branched covers over the trefoil

“…. Therefore M is the prism manifold M(3, 2) in the notation of [23]. Compare it with the 4-fold cyclic covering depicted in Figure 5(c).…”

“…Hence, M 3 is a prism manifold, which are characterized by their fundamental group. According to the notation of [23] this is M (2, 1), also called the Quaternionic Space [17]. Consider now the presentation m → (1, 2, 3, 4), c → (1, 4, 3, 2), the fundamental group of M 4 is SL 2 (Z 3 ) ∼ = Z 3 Q 8 .…”

Higher Spins from Nambu–Chern–Simons Theory

“…If we remove the volume-preservation restriction, it is certainly true that fibre-preserving transformations project down to the full diffeomorphism group of the S 2 base (this is the content of the "Projection theorem" of Cerf and Palais, see e.g. [37]). Whether this is still true if we restrict to fibre-preserving, total-space-volume-preserving diffeomorphisms is apparently unknown for nontrivial bundles but is clearly true for trivial ones.…”