Forme de contact sur la somme connexe de deux variétés de contact de dimension impaire

Meckert, Christiane

doi:10.5802/aif.888

Cited by 19 publications

(17 citation statements)

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“…This is done by connect summing. It is well known that the connected sum of two contact manifolds carries again a contact structure, see [11] and [16]. In his thesis Ustilovsky showed that we can roughly see what happens to the resulting contact homology, provided we are given sufficient information on the given two contact manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…Given two contact manifolds (M 1 , ξ 1 ) and (M 2 , ξ 2 ), we can build a new contact manifold by forming their connected sum, [11] and [16]. If we think of a connected sum as first removing a disk from both M 1 and M 2 and then gluing them via a "connecting" tube, then the contact structure on M 1 #M 2 can be be made to coincide with the contact structure on M 1 with a disk removed and M 2 with a disk removed.…”

Section: Exotic Contact Structuresmentioning

confidence: 99%

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Contact homology of Brieskorn manifolds

Koert¹

2008

Forum Mathematicum

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Abstract. We give an algorithm for computing the contact homology of some Brieskorn manifolds. Brieskorn manifolds can be regarded as circle-bundles over orbifolds and our algorithm expresses cylindrical contact homology of the Brieskorn manifold in terms of the homology of the underlying orbifold. As an application, we construct infinitely many contact structures on the class of simply connected contact manifolds that admit nice contact forms (i.e. no Reeb orbits of degree -1, 0 or 1) and have index positivity with trivial first Chern class. IntroductionFor a long time it has been known that contact manifolds may carry many nonisomorphic contact structures. A first way to distinguish these structures from each other is by considering their Chern class or their formal homotopy class. In dimension 3 we can, in addition, sometimes distinguish contact structures by showing tightness or overtwistedness as was shown first by Bennequin. At present, the latter two notions do not have generalizations to higher dimensions.Eliashberg was the first to show that the classical invariants (i.e. the Chern class and formal homotopy class) are not always enough for a full classification of contact structures on a given manifold in dimension larger than 3. He showed that spheres in dimensions 4n + 1 admit a non-standard, yet homotopically trivial contact structure in [4]. Giroux gave an interesting example in dimension 3 in [7]. He exhibited infinitely many tight contact structures on T 3 in the same homotopy class of plane fields. Since then more general techniques to distinguish contact structures were introduced. Eliashberg and Hofer's contact homology is such a technique (for a survey, see [5], and for a more recent description, [6]). It works, roughly speaking, as follows. The Reeb orbits on a given contact manifold can be given a grading. The chain complexes are freely generated by the closed Reeb orbits. Then a differential can be defined by counting certain holomorphic curves asymptotic to Reeb orbits in the symplectization of the contact manifold. It can be shown that the homology of this differential is independent of the contact form under suitable conditions and it is hence an invariant of the contact structure. Since its introduction, several examples of non-isomorphic contact structures in the same homotopy class have been found.One of these examples is a family of Brieskorn spheres in dimensions 4n + 1. These were found by Ustilovsky in [14]. In his thesis he used a few other tools that can easily be applied to construct more contact structures on the same manifold. This is done by connect summing. It is well known that the connected sum of two contact manifolds carries again a contact structure, see [11] and [16]. In his thesis 1991 Mathematics Subject Classification. Primary 53D10 .

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Section: Introductionmentioning

confidence: 99%

Section: Exotic Contact Structuresmentioning

confidence: 99%

Contact homology of Brieskorn manifolds

Koert¹

2008

Forum Mathematicum

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“…Let (M, ξ) be a contact 3-manifold with contact structure induced by a nondegenerate contact form λ, and let J ∈ J (M, λ) be a complex multiplication for which the triple (M, λ, J) admits a stable finite energy foliation F of energy E(F). Then, there exists an open, dense set U ⊂ M × M \ ∆(M ) so that for any (p, q) ∈ U the contact manifold (M , ξ ) obtained by performing a contact connected sum at (p, q) as in [41,54] admits a nondegenerate contact form λ with ξ = ker λ , a compatible J ∈ J (M , λ ) and a stable finite energy foliation F for the data (M , λ , J ) with energy E(F ) = E(F).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Connected sums and finite energy foliations I: Contact connected sums

Fish

Siefring

2018

Journal of Symplectic Geometry

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We consider a 3-manifold M equipped with nondegenerate contact form λ and compatible almost complex structure J. We show that if the data (M, λ, J) admits a stable finite energy foliation, then for a generic choice of distinct points p, q ∈ M , the manifold M formed by taking the contact connected sum at p and q admits a nondegenerate contact form λ and compatible almost complex structure J so that the data (M , λ , J ) also admits a stable finite energy foliation. Along the way, we develop some general theory for the study of finite energy foliations.

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“…Hence by a theorem of E. Giroux [10] S n × S m is a contact manifold. Now, it was shown by C. Meckert [22] and more generally by Weinstein [25] (see also [9]) that the connected sum of contact manifolds of the same dimension is a contact manifold. Therefore all odd dimensional connected sums of sphere products admit contact structures.…”

Section: Theorem 4 If K Is Even Zmentioning

confidence: 97%

Some open book and contact structures on moment-angle manifolds

Barreto

Medrano

Verjovsky

2016

Bol. Soc. Mat. Mex.

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I.We construct open book structures on all moment-angle manifolds and describe the topology of their leaves and bindings under certain restrictions. II. We also show, using a recent deep result about contact forms due to Borman, Eliashberg and Murphy [6], that every odd-dimensional moment-angle manifold admits a contact structure. This contrasts with the fact that, except for a few, well-determined cases, even-dimensional ones do not admit symplectic structures. We obtain the same results for large families of more general intersections of quadrics.

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Forme de contact sur la somme connexe de deux variétés de contact de dimension impaire

Cited by 19 publications

References 1 publication

Contact homology of Brieskorn manifolds

Contact homology of Brieskorn manifolds

Connected sums and finite energy foliations I: Contact connected sums

Some open book and contact structures on moment-angle manifolds

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