Geometry & Topology, Vol. 9 (2005) Paper no. 50, pages 2193--2226.

Constructions contrôlées de champs de Reeb et applications

Vincent Colin, Ko Honda

Abstract. On every compact, orientable, irreducible 3-manifold V which is toroidal or has torus boundary components we construct a contact 1-form whose Reeb vector field R does not have any contractible periodic orbits and is tangent to the boundary. Moreover, if bdry V is nonempty, then the Reeb vector field R is transverse to a taut foliation. By appealing to results of Hofer, Wysocki, and Zehnder, we show that, under certain conditions, the 3-manifold obtained by Dehn filling along bdry V is irreducible and different from the 3-sphere.

Résumé. On construit, sur toute variete V de dimension trois orientable, compacte, irreductible, bordee par des tores ou toroidale, une forme de contact dont le champ de Reeb R est sans orbite periodique contractible et tangent au bord. De plus, si le bord de V est non vide, le champ R est transversal a un feuilletage tendu. En utilisant des resultats de Hofer, Wysocki et Zehnder, on obtient sous certaines conditions que la variete obtenue par obturation de Dehn le long du bord de V est irreductible et differente de la sphere S^3.

Keywords. Reeb vector field, contact structure, taut foliation

AMS subject classification. Primary: 53D35. Secondary: 53C15.

E-print: arXiv:math.GT/0411640

DOI: 10.2140/gt.2005.9.2193

Submitted to GT on 25 November 2004. (Revised 4 September 2005.) Paper accepted 26 November 2005. Paper published 1 December 2005.

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Vincent Colin, Ko Honda
Universite de Nantes, UMR 6629 du CNRS, 44322 Nantes, France
and
University of Southern California, Los Angeles, CA 90089, USA
Email: Vincent.Colin@math.univ-nantes.fr, khonda@math.usc.edu
URL: http://rcf.usc.edu/~khonda

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