Convexité en topologie de contact

Giroux, Emmanuel

doi:10.1007/bf02566670

Cited by 309 publications

(455 citation statements)

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“…Concerning the second fact, we will always use in this note the conventions of Giroux [11] on orientations. In particular, a singularity p of F ξ (P ) has positive divergence if and only if the orientations of ξ and P coincide at p.…”

Section: Branched Standard Polyhedra Embeddings and Foliationsmentioning

confidence: 99%

“…We can also assume that ξ t is defined on a neighbourhood V t of the closure of U t , and that the foliation induced by ξ t on S t is the trivial one, with one source and one sink and no saddles or cycles. The last condition can be imposed to S 0 according to [11], and is then automatic for all S t 's. Now we can extend each ξ t to M by identifying the ball M \ U t with the unit ball in the standard tight structure on R 3 .…”

Section: 1mentioning

confidence: 99%

“…In particular (see [10]) one can describe exactly what foliations arise ("existence results"). Moreover one knows that the characteristic foliation F = F ξ (Σ) determines uniquely the contact structure ξ in a neighbourhood of a surface Σ ("uniqueness results").…”

mentioning

confidence: 99%

“…(By convention in this note all surfaces are closed and oriented, all 3-manifolds are oriented, all plane fields are cooriented and all contact structures are positive.) Moreover if F is generic enough, so that it admits a splitting curve Γ (see [10]) the geometry of ξ is deeply related to the topology of the splitting; in particular Γ determines whether ξ is tight or not near Σ ( [11], [12]). One of the basic tools in this subject is the so-called elimination lemma (see [10], [6]).…”

mentioning

confidence: 99%

“…To establish fact 2 we use Eliashberg's [6] uniqueness theorem for tight structures on the ball. Facts 3 and 5 extend the proofs of Giroux [10] using ad hoc arguments to deal with singularities, while 4 uses the genuine (unbranched) elimination lemma.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Branched spines and contact structures on 3-manifolds

Benedetti

Petronio

2000

Annali di Matematica pura ed applicata

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Section: Branched Standard Polyhedra Embeddings and Foliationsmentioning

confidence: 99%

Section: 1mentioning

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confidence: 99%

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Branched spines and contact structures on 3-manifolds

Benedetti

Petronio

2000

Annali di Matematica pura ed applicata

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A characterization of the tight 3-sphere II

Hofer¹,

Wysocki²,

Zehnder³

1999

Comm. Pure Appl. Math.

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3-Dimensional Methods in Contact Geometry

Honda

International Mathematical Series

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Introduction 1.1. First examples 1.2. Legendrian knots 1.3. Tight vs. overtwisted 1.4. Classification of contact structures 1.5. A criterion for tightness 1.6. Relationship with foliation theory 2. Convex surfaces 2.1. Characteristic foliations 2.2. Convexity 2.3. Convex decomposition theory 3. Bypasses 3.1. Definition and examples 3.2. Finding bypasses 4. Classification of tight contact structures on lens spaces 4.1. The standard neighborhood of a Legendrian curve 4.2. Lens spaces 4.3. Solid tori 4.4. Completion of the proof of Theorems 4.2 and 4.3 5. Gluing 5.1. Basic examples with trivial state transitions 5.2. More complicated example 5.3 . Tightness and fillability ReferencesA contact manifold (M, ξ) is a (2n + 1)-dimensional manifold M equipped with a smooth maximally nonintegrable hyperplane field ξ ⊂ T M, i.e., locally ξ = ker α, where α is a 1-form which satisfies α ∧ (dα) n = 0. Since dα is a nondegenerate 2-form when restricted to ξ, contact geometry is customarily viewed as the odd-dimensional sibling of symplectic geometry. Although Date: December 31, 2003. 1 2 KO HONDAcontact geometry in dimensions ≥ 5 is still in an incipient state, contact structures in dimension 3 are much better understood, largely due to the fact that symplectic geometry in two dimensions is just the study of area. The goal of this article is to explain some of the recent developments in 3-dimensional contact geometry, with an emphasis on methods from 3-dimensional topology. Basic references include [Ae, El2, Et1, Ge]. The article [Kz] is similar in spirit to ours.Three-dimensional contact geometry lies at the interface between 3-and 4-manifold geometries, and has been an essential part of the flurry in low-dimensional geometry and topology over the last 20 years. In dimension 3, it relates to foliation theory and knot theory; in dimension 4, there are rich interactions with symplectic geometry. In both dimensions, there are relations with gauge theories such as Seiberg-Witten theory and Heegaard Floer homology. Acknowledgements. This manuscript grew out of a lecture series given at the Winter School in Contact Geometry in München in February 2003 and a minicourse given at the Geometry and Foliations 2003 conference, held at Ryokoku University in Kyoto in September 2003. I would like to thank Kai Cieliebak and Dieter Kotschick for the former, and Takashi Tsuboi for the latter, as well as for his hospitality during my visit to the University of Tokyo and the Tokyo Institute of Technology during the summer and fall of 2003. Much of the actual writing took place during this visit.

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Convexité en topologie de contact

Cited by 309 publications

References 8 publications

Branched spines and contact structures on 3-manifolds

Branched spines and contact structures on 3-manifolds

A characterization of the tight 3-sphere II

3-Dimensional Methods in Contact Geometry

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