Rigidity versus flexibility for tight confoliations

Vogel, Thomas

doi:10.2140/gt.2011.15.41

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Rigidity versus flexibility for tight confoliations

Thomas Vogel

Geometry & Topology 15 (2011) 41–121

DOI: 10.2140/gt.2011.15.41

Abstract

In their book "Confoliations" [Univ. Lecture Ser. 13, Amer. Math. Soc. (1998)], Y Eliashberg and W Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $ξ$ on $T^{3}$ violating the Thurston–Bennequin inequalities. This answers a question from "Confoliations" negatively. Despite this, it is still possible to prove restrictions on homotopy classes of plane fields which contain tight confoliations.

The failure of the Thurston–Bennequin inequalities for tight confoliations is due to the presence of overtwisted stars. Overtwisted stars are particular configurations of Legendrian curves which bound a disc with finitely many punctures on the boundary. We prove that the Thurston–Bennequin inequalities hold for tight confoliations without overtwisted stars and that symplectically fillable confoliations do not admit overtwisted stars.

Keywords

tight, confoliation

Mathematical Subject Classification 2000

Primary: 57R17, 57R30

References

Publication

Received: 30 March 2009

Revised: 16 June 2010

Accepted: 18 October 2010

Published: 6 January 2011

Proposed: Yasha Eliashberg

Seconded: Danny Calegari, Leonid Polterovich

Authors

Thomas Vogel
Max-Planck-Institut für Mathematik Vivatsgasse 7 D-53129 Bonn Germany

Volume 15, issue 1 (2011)