Comparing Heegaard splittings of non-haken 3-manifolds

Rubinstein, Hyam; Scharlemann, Martin

doi:10.1016/0040-9383(95)00055-0

Cited by 82 publications

(167 citation statements)

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“…(And presumably for 2-bridge links as well, though we do not pursue that here, because of the technical obstacle that the theory in [STo] so far has not been explicitly extended to 3-manifolds with non-empty boundary. Compare [RS2] to [RS1].) This result can viewed as the analogue for bridge surfaces of the result of Bonahon and Otal mentioned above.…”

Section: Introductionmentioning

confidence: 87%

“…This result can viewed as the analogue for bridge surfaces of the result of Bonahon and Otal mentioned above. Our approach will be analogous to that of [RS1], working from the central result of [STo]: in the absence of incompressible Conway spheres, two c-weakly incompressible bridge surfaces can be properly isotoped to intersect in a non-empty collection of closed curves, each of which is essential (including non-meridional) in both surfaces.…”

Section: Introductionmentioning

confidence: 99%

“…In [BoO], Bonahon and Otal proved that the same is true of lens splaces (manifolds with a genus one Heegaard surface). A later proof [RS1] made use of the fact that any two weakly incompressible Heegaard splittings of a manifold can be isotoped to intersect in a nonempty collection of curves that are essential on both Heegaard surfaces.…”

Section: Introductionmentioning

confidence: 99%

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Uniqueness of bridge surfaces for 2-bridge knots

Scharlemann

Tomova

2008

Math. Proc. Camb. Phil. Soc.

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Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

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Uniqueness of bridge surfaces for 2-bridge knots

Scharlemann

Tomova

2008

Math. Proc. Camb. Phil. Soc.

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“…In [18], Rubinstein-Scharlemann introduced a powerful machinery, which is called a graphic, for studying Heegaard splittings of 3-manifolds, and succeeded to obtain deep results on the Reidemeister-Singer distance of two strongly irreducible Heegaard splittings of a 3-manifold. We note that Rubinstein and Scharlemann derived a graphic from two Heegaard splittings of a 3-manifold via Cerf theory [5].…”

Section: Introductionmentioning

confidence: 99%

“…In [19], Rubinstein and Scharlemann give a generalization of results in [18] for 3-manifolds with boundary. As the second application, we will give another formulation for generalizing the idea in [18] for link spaces. In fact, we will introduce an orbifold version of the Rubinstein-Scharlemann type argument(Sect.…”

Section: Introductionmentioning

confidence: 99%

The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set of a stable map

Kobayashi

Saeki

2000

Pacific J. Math.

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We show that Rubinstein-Scharlemann graphics for 3-manifolds can be regarded as the images of the singular sets (: discriminant set) of stable maps from the 3-manifolds into the plane. As applications of our understanding of the graphic, we give a method for describing Heegaard surfaces in 3-manifolds by using arcs in the plane, and give an orbifold version of Rubinstein-Scharlemann's setting. Then by using this setting, we show that every genus one 1-bridge position of a nontrivial two bridge knot is obtained from a 2-bridge position in a standard manner.

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Diffeomorphisms of Elliptic 3-Manifolds

Hong

Kalliongis

McCullough

et al. 2012

Lecture Notes in Mathematics

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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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Comparing Heegaard splittings of non-haken 3-manifolds

Cited by 82 publications

References 7 publications

Uniqueness of bridge surfaces for 2-bridge knots

Uniqueness of bridge surfaces for 2-bridge knots

The Rubinstein–Scharlemann graphic of a 3-manifold as the discriminant set of a stable map

Diffeomorphisms of Elliptic 3-Manifolds

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