Scindements de Heegaard et groupe des hom�otopies des petites vari�t�s de Seifert

Boileau, Michel; Otal, Jean-Pierre

doi:10.1007/bf01243906

Cited by 40 publications

(81 citation statements)

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“…In Section 3, we determine which lens spaces have property H. In Section 4, we compute Out. 1 .M // by combinatorial methods. Mapping class groups of S 3 -manifolds are computed in Section 5.…”

Section: Resultsmentioning

confidence: 99%

“…Mapping class groups of S 3 -manifolds are computed in Section 5. Although the mapping class groups of S 3 -manifolds are determined by Boileau and Otal [1] and McCullough [12] and some partial results are given by Hodgson and Rubinstein [8], Rubinstein [17] and Rubinstein and Birman [18], we give a complete computation based on the fact that all self-homeomorphisms on an S 3 -manifold M ¤ L.p; q/ can be isotopic to a fiber-preserving homeomorphism. In Section 6, Table 1 shows the computation results.…”

Section: Resultsmentioning

confidence: 99%

“…where N is a S 3 -manifold, 1 .N / belongs to one of the previous seven types and j 1 .N /j is coprime to m. The cyclic group Z p is realized by lens space L.p; q/. Each group in the remaining types is realized by a unique S 3 -manifold.…”

Section: Resultsmentioning

confidence: 99%

“…1 .M // and the mapping class group of M . In Section 3, we determine which lens spaces have property H. In Section 4, we compute Out.…”

Section: Resultsmentioning

confidence: 99%

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Degree ±1 self-maps and self-homeomorphisms on prime 3-manifolds

Sun

2010

Algebr. Geom. Topol.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

“…1 .M // and the mapping class group of M . In Section 3, we determine which lens spaces have property H. In Section 4, we compute Out.…”

Section: Resultsmentioning

confidence: 99%

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Degree ±1 self-maps and self-homeomorphisms on prime 3-manifolds

Sun

2010

Algebr. Geom. Topol.

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“…This property is known to be true for all Seifert fiber spaces which are not covered by S 2 x S 1 , S s or T 3 [Sc,Theorem 3.9], as well as for those manifolds which have a sphere orbifold with cone points of order (2,3,p), p > 5 or (3,3, fc), k > 2 [BO,Proposition 3.1]. In Section 3 we will show that this is true for all but four Seifert fiber spaces.…”

Section: Theorem 14 Suppose X Is a Hyperbolic Orbifold Then Any Homentioning

confidence: 99%

Homeomorphisms of 3-manifolds and the realization of Nielsen number

Jiang¹,

Wang²,

Wu³

2001

Communications in Analysis and Geometry

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The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f , which is a lower bound for the number of fixed points among all maps homotopic to f . The main theorem of this paper proves this conjecture for all orientation preserving maps on geometric or Haken 3-manifolds. It will also be shown that on many manifolds all maps are isotopic to fixed point free maps.The proof is based on the understanding of homeomorphisms on 2-orbifolds and 3-manifolds. Thurston's classification of surface homeomorphisms will be generalized to 2-dimensional orbifolds, which is used to study fiber preserving maps of Seifert fiber spaces. Maps on most Seifert fiber spaces are indeed isotopic to fiber preserving maps, with the exception of four manifolds and orientation reversing maps on lens spaces or S 3 . It will also be determined exactly which manifolds have a unique Seifert fibration up to isotopy. These informations will be used to deform a map to certain standard map on each piece of the JSJ decomposition, as well as on the neighborhood of the decomposition tori, which will make it possible to shrink each fixed point class to a single point, and remove inessential fixed point classes.

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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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Scindements de Heegaard et groupe des hom�otopies des petites vari�t�s de Seifert

Cited by 40 publications

References 31 publications

Degree ±1 self-maps and self-homeomorphisms on prime 3-manifolds

Degree ±1 self-maps and self-homeomorphisms on prime 3-manifolds

Homeomorphisms of 3-manifolds and the realization of Nielsen number

Diffeomorphisms of Elliptic 3-Manifolds

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