Classification of overtwisted contact structures on 3-manifolds

Eliashberg, Yakov

doi:10.1007/bf01393840

Cited by 348 publications

(430 citation statements)

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“…In particular, by combining this statement with Theorem 1.1, one recovers a result first proved by Eliashberg using different methods, namely the statement that overtwisted contact structures are not fillable [9]. The vanishing also echoes another result due to Eliashberg [6], which states that the classification of overtwisted contact structures is essentially soft: it is the same as the homotopy classification of 2-plane fields on the 3-manifold.…”

Section: (Iii) Further Propertiessupporting

confidence: 72%

Monopoles and contact structures

Kronheimer

Mrowka

1997

Inventiones Mathematicae

133

264

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Section: (Iii) Further Propertiessupporting

confidence: 72%

Monopoles and contact structures

Kronheimer

Mrowka

1997

Inventiones Mathematicae

133

264

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“…10 A surface S inside a contact 3-manifold determines a singular foliation on S, called the characteristic foliation of S, by the intersection of the contact planes with the tangent spaces to S. A contact structure on a 3-manifold M is called overtwisted if there exists an embedded 2-disk whose characteristic foliation contains one closed leaf C and exactly one singular point inside C; otherwise, the contact structure is called tight. Eliashberg [32] showed that the isotopy classification of overtwisted contact structures on closed 3-manifolds coincides with their homotopy classification as tangent plane fields. The classification of tight contact structures is still open.…”

Section: Symplectizationmentioning

confidence: 98%

Lectures on Symplectic Geometry

2001

Lecture Notes in Mathematics

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ForewordThese notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997.The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to symplectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT informal symplectic seminar 93-96, I would References 199 Index 207 IntroductionThe goal of these notes is to provide a fast introduction to symplectic geometry. A symplectic form is a closed nondegenerate 2-form. A symplectic manifold is a manifold equipped with a symplectic form. Symplectic geometry is the geometry of symplectic manifolds. Symplectic manifolds are necessarily even-dimensional and orientable, since nondegeneracy says that the top exterior power of a symplectic form is a volume form. The closedness condition is a natural differential equation, which forces all symplectic manifolds to being locally indistinguishable. (These assertions will be explained in Lecture 1 and Homework 2.)The list of questions on symplectic forms begins with those of existence and uniqueness on a given manifold. For specific symplectic manifolds, one would like to understand the geometry and the topology of special submanifolds, the dynamics of certain vector fields or systems of differential equations, the symmetries and extra structure, etc.Two centuries ago, symplectic geometry provided a language for classical mechanics. Through its recent huge development, it conquered an independent and rich territory, as a central branch of differential geometry and topology. To mention just a few key landmarks, one may say that symplectic geometry began to take its modern shape with the formulation of the Arnold conjectures in the 60's and with the foundational work of Weinstein in the 70's. A paper of Gromov [49] in the 80's gave the subject a whole new set of tools: pseudo-holomorphic curves. Gromov also first showed that important results from complex Kähler geometry remain true in the more general symplectic category, and this direction was continued rather dramatically in the 90's in the work of Donaldson on the topology of symplectic manifolds and their symplectic submanifolds, and in the work of Taubes in the context of the Seiberg-Witten invariants. Symplectic geometry is significantly stimulated by important interactions with global analysis, mathematical physics, low-dimensional topology, dynamical systems, algebraic geometry, integrable systems, microlocal analysis, partial differential equations, representation theory, quantization, equivariant cohomology, geometric combinatorics, etc.As a curiosity, note that two centuries ago the name symplectic geometry did not exist. If you consult a major English dictionary,...

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“…It was understood in 1989 (see [23]) that in the world of 3-dimensional contact manifolds there is an important dichotomy. If a contact manifold contains the so-called overtwisted disc, i.e., an embedded disc which along its boundary is tangent to the contact structure, then the contact structure becomes very flexible and abides by a certain h-principle: two overtwisted contact structures which are homotopic as plane fields are homotopic as contact structures, and hence in view of Gray's theorem are isotopic.…”

Section: Flexible Milestones After the Resolution Of Gromov's Alternamentioning

confidence: 99%

Untitled

2015

Bull. Amer. Math. Soc.

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Abstract. Flexible and rigid methods coexisted in symplectic topology from its inception. While the rigid methods dominated the development of the subject during the last three decades, the balance has somewhat shifted to the flexible side in the last three years. In the talk we survey the recent advances in symplectic flexibility in the work of S. Borman, K. Cieliebak, T. Ekholm, E. Murphy, I. Smith, and the author.This survey article is an expanded version of author's article [27] and his talk at the Current Events Bulletin at the AMS meeting in Baltimore in January 2014. It also uses materials from the papers [6,9,20,32] and the book [8]. The h-principleMany problems in mathematics and its applications deal with partial differential equations, partial differential inequalities or, more generally, with partial differential relations (see [48] and also [31]), i.e., any conditions imposed on partial derivatives of an unknown function. A solution of such a partial differential relation R is any function which satisfies this relation.With any differential relation one can associate the underlying algebraic relation by substituting all the derivatives entering the relation with new independent functions. The existence of a solution of the corresponding algebraic relation, called a formal solution of the original differential relation R, is a necessary condition for the solvability of R. Though it seems that this necessary condition should be very far from being sufficient, it was a surprising discovery in the 1950s of geometrically interesting problems where existence of a formal solution is the only obstruction for the genuine solvability. Two of the first such non-trivial examples were the C 1 -isometric embedding theorem of J. Nash and N. Kuiper [57,68] and the immersion theory of S. Smale and M. Hirsch [55,77]. After Gromov's remarkable series of papers, beginning with his paper [47] and culminating in his book [48], the area crystallized as an independent subject, called the h-principle.Rigid and flexible results coexist in many areas of geometry, but nowhere else do they come so close to each other, as in symplectic topology, which serves as a rich source of examples on both sides of the flexible-rigid spectrum. Flexible and rigid problems and the development of each side toward the other shaped and continue to shape the subject of symplectic topology from its inception.

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Classification of overtwisted contact structures on 3-manifolds

Cited by 348 publications

References 5 publications

Monopoles and contact structures

Monopoles and contact structures

Lectures on Symplectic Geometry

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