Volume 9, issue 4 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Constructions controlees de champs de Reeb et applications

Vincent Colin and Ko Honda

Geometry & Topology 9 (2005) 2193–2226

arXiv: math.GT/0411640

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 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 V is irreducible and different from the 3–sphere.

Résumé

On construit, sur toute variété V de dimension trois orientable, compacte, irréductible, bordée par des tores ou toroïdale, une forme de contact dont le champ de Reeb R est sans orbite périodique contractible et tangent au bord. De plus, si V est non vide, le champ R est transversal à un feuilletage tendu. En utilisant des résultats de Hofer, Wysocki et Zehnder, on obtient sous certaines conditions que la variété obtenue par obturation de Dehn le long du bord de V est irréductible et différente de la sphère S3.

Keywords
Reeb vector field, contact structure, taut foliation
Mathematical Subject Classification 2000
Primary: 53D35
Secondary: 53C15
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Publication
Received: 25 November 2004
Revised: 4 September 2005
Accepted: 26 November 2005
Published: 1 December 2005
Proposed: Yasha Eliashberg
Seconded: Tomasz Mrowka, Joan Birman
Authors
Vincent Colin
Université de Nantes
UMR 6629 du CNRS
44322 Nantes
France
Ko Honda
University of Southern California
Los Angeles
California 90089
USA