Algebraic & Geometric Topology

Paires de structures de contact sur les variétés de dimension trois

Vincent Colin and Sebastião Firmo

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Abstract

On introduit une notion de paire positive de structures de contact sur les variétés de dimension trois qui généralise celle de Eliashberg et Thurston [Confoliations, Univ. Lecture Series 13, Amer. Math. Soc. (1998)] et Mitsumatsu [Ann. Inst. Fourier (Grenoble) 45 (1995) 1407–1421 ; Foliations : geometry and dynamics (Warsaw, 2000) World Sci. Publ., River Edge, NJ (2002) 75–125]. Une telle paire “normale” donne naissance à un champ de plans continu et localement intégrable λ. On montre que si λ est uniquement intégrable et si les structures de contact sont tendues, alors le feuilletage intégral de λ est sans composante de Reeb d’âme homologue à zéro. De plus, dans ce cas, la variété ambiante porte un feuilletage sans composante de Reeb. On démontre également un théorème de stabilité “à la Reeb” pour les paires positives de structures tendues.

We introduce the notion of a positive pair of contact structures of a three dimensional manifold which generalises that of Eliashberg, Thurston and Mitsumatsu. A normal such pair gives rise to a continuous, locally integrable plane field λ. We show that if lambda is uniquely integrable and if the contact structures are tight then the integral foliation of λ has no Reeb component whose core is homologous to zero. Moreover, in this case, the ambient manifold carries a foliation without a Reeb component. We also show a Reeb stability theorem for positive pairs of tight structures.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 5 (2011), 2627-2653.

Dates
Received: 2 June 2009
Accepted: 25 July 2011
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715300

Digital Object Identifier
doi:10.2140/agt.2011.11.2627

Mathematical Reviews number (MathSciNet)
MR2836297

Zentralblatt MATH identifier
1234.57019

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology 57R30: Foliations; geometric theory

Keywords
structure de contact pair of contact structure paire feuilletage foliation tendu tight composante de Reeb reeb component

Citation

Colin, Vincent; Firmo, Sebastião. Paires de structures de contact sur les variétés de dimension trois. Algebr. Geom. Topol. 11 (2011), no. 5, 2627--2653. doi:10.2140/agt.2011.11.2627. https://projecteuclid.org/euclid.agt/1513715300


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