Pinwheels and bypasses. (English) Zbl 1086.57016

This paper builds upon earlier work by the authors concerning tight contact structures on 3–manifolds. Here they consider the question as to when the attachment of a family of bypasses along a family of Legendrian curves to a convex surface results in a tight contact manifold. (The attachment of bypasses was introduced by K. Honda in [Geom. Topol. 4, 309–368 (2000; Zbl 0980.57010)]). The answer is that a necessary and sufficient condition for this to happen is the nonexistence of a polygonal region which they call a virtual pinwheel.


57M50 General geometric structures on low-dimensional manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension


Zbl 0980.57010
Full Text: DOI arXiv EuDML EMIS


[1] V Colin, Une infinité de structures de contact tendues sur les variétés toroïdales, Comment. Math. Helv. 76 (2001) 353 · Zbl 1005.57014
[2] Y Eliashberg, Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165 · Zbl 0756.53017
[3] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637 · Zbl 0766.53028
[4] E Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001) 218 · Zbl 0988.57015
[5] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309 · Zbl 0980.57010
[6] K Honda, W H Kazez, G Matić, Tight contact structures and taut foliations, Geom. Topol. 4 (2000) 219 · Zbl 0961.57009
[7] K Honda, W H Kazez, G Matić, Convex decomposition theory, Int. Math. Res. Not. (2002) 55 · Zbl 0992.57008
[8] K Honda, W H Kazez, G Matić, Tight contact structures on fibered hyperbolic 3-manifolds, J. Differential Geom. 64 (2003) 305 · Zbl 1083.53082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.