Géométrie de contact: de la dimension trois vers les dimensions supérieures

By Emmanuel Giroux. 2003

529 citations recorded by the source corpus.

On d\'ecrit ici des relations entre la g\'eom\'etrie globale des vari\'et\'es de contact closes et celle de certaines vari\'et\'es symplectiques, \`a savoir les vari\'et\'es de Stein compactes. L'origine de ces relations est l'existence de livres ouverts adapt\'es aux structures de contact. We discuss relations between the global geometry of closed contact manifolds and the geometry of compact symplectic Stein manifolds that they bound. The origin of these relations is the existence of open book decompositions adapted to contact structures.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/877e2e09f49922e45878bb95704d531d80421d86

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Braids: A Survey (2004, background): “In 2003 Giroux proved [79] that every contact structure on every closed, orientable 3-manifold M3 can be obtained in the following way: represent M3 as a branched covering space of S3, branched over a knot or link, and lift the standard and overtwisted contact structures on S3 to M3.”
Knot Theory of Complex Plane Curves (2004, background): “ably positive Hopf-plumbed in case some positive Hopfplumbed ﬁber surface Fcan also be constructed as an iterated Murasugi sum of positive Hopf annuli A(O,−1) starting from S. A remarkable theorem of Giroux (2002) states that a ﬁbered link Lhas a stably positive Hopf-plumbed ﬁber surface if and only”
Symplectic hypersurfaces and transversality in Gromov-Witten theory (2007, unknown): “The extension to genus zero holomorphic curves with boundary on a Lagrangian submanifold as in Floer homology [11] relies on the existence of approximately holomorphic hypersurfaces in the complement of a Lagrangian submanifold due to Auroux, Gayet and Mohsen [3], and the extension to genus zero hol”
Essential open book foliations and fractional Dehn twist coefficient (2012, background): “Due to the Giroux-correspondence the open book foliation has extensive application to contact geometry.”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “Giroux proved that the map which sends an open book decomposition to a compatible contact structure gives rise to a one-to-one correspondence from set of open book decompositions of Y up to positive stabilization to the set of contact structures on Y up to isotopy [Gir02].”
Altering symplectic manifolds by homologous recombination (2010, background): “From now on, whenever we discuss handle attachment, these additional diﬀeomorphism data are assumed to be present. Weinstein handles appear in the context of stabilization of open book decompositions [28], and also in the related argument for Lefschetz ﬁbrations. Suppose as before that M0 contains a”
THE WEINSTEIN CONJECTURE FOR PLANAR CONTACT STRUCTURES IN DIMENSION THREE (2004, unknown): “The deﬁnition implies that each component of L is a periodic orbit of the Reeb vector ﬁeld associated to the Giroux form.”
Lectures on open book decompositions and contact structures (2004, background): “ parameterized curve (f(r),g(r)).This curve is deﬁned for rnear 0 and 1; can we extend it over all of [0,1] so that the position and tangent vector are never collinear? Proposition 3.18 (Giroux 2000, [21]). Two contact structures supported by the same open book are isotopic. Proof. Let α0 and α1 be ”
Geometric criteria for overtwistedness (2015, background): “Giroux in his study of the correspondance between open books and contact structures [17, 31].”
A biased view of symplectic cohomology (2007, background): “This should give rise to an open book decomposition of ∂M , compatible with its contact structure in the sense of [20].”