Geodesic
A class of tight contact structures on Σ2 × I
Cofer, Tanya1
1Department of Mathematics, Northeastern Illinois University, 5500 North St Louis Avenue, Chicago IL 60625-4699, USA
Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 961-1011
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus–2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case.

DOI : 10.2140/agt.2004.4.961
Keywords: tight, contact structure, genus-2 surface

Cofer, Tanya  1

1 Department of Mathematics, Northeastern Illinois University, 5500 North St Louis Avenue, Chicago IL 60625-4699, USA 
@article{10_2140_agt_2004_4_961,
     author = {Cofer, Tanya},
     title = {A class of tight contact structures on {\ensuremath{\Sigma}2} {\texttimes} {I}},
     journal = {Algebraic and Geometric Topology},
     pages = {961--1011},
     year = {2004},
     volume = {4},
     number = {2},
     doi = {10.2140/agt.2004.4.961},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.961/}
}
TY  - JOUR
AU  - Cofer, Tanya
TI  - A class of tight contact structures on Σ2 × I
JO  - Algebraic and Geometric Topology
PY  - 2004
SP  - 961
EP  - 1011
VL  - 4
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.961/
DO  - 10.2140/agt.2004.4.961
ID  - 10_2140_agt_2004_4_961
ER  - 
%0 Journal Article
%A Cofer, Tanya
%T A class of tight contact structures on Σ2 × I
%J Algebraic and Geometric Topology
%D 2004
%P 961-1011
%V 4
%N 2
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2004.4.961/
%R 10.2140/agt.2004.4.961
%F 10_2140_agt_2004_4_961
Cofer, Tanya. A class of tight contact structures on Σ2 × I. Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 961-1011. doi: 10.2140/agt.2004.4.961

[1] D Bennequin, Entrelacements et équations de Pfaff, from: "Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)", Astérisque 107, Soc. Math. France (1983) 87

[2] V Colin, Recollement de variétés de contact tendues, Bull. Soc. Math. France 127 (1999) 43

[3] V Colin, Une infinité de structures de contact tendues sur les variétés toroïdales, Comment. Math. Helv. 76 (2001) 353

[4] V Colin, E Giroux, K Honda, On the coarse classification of tight contact structures, from: "Topology and geometry of manifolds (Athens, GA, 2001)", Proc. Sympos. Pure Math. 71, Amer. Math. Soc. (2003) 109

[5] Y Eliashberg, Classification of overtwisted contact structures on 3–manifolds, Invent. Math. 98 (1989) 623

[6] Y Eliashberg, Contact 3–manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992) 165

[7] J B Etnyre, Tight contact structures on lens spaces, Commun. Contemp. Math. 2 (2000) 559

[8] J B Etnyre, K Honda, On the nonexistence of tight contact structures, Ann. of Math. $(2)$ 153 (2001) 749

[9] E Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991) 637

[10] E Giroux, Structures de contact en dimension trois et bifurcations des feuilletages de surfaces, Invent. Math. 141 (2000) 615

[11] E Giroux, Structures de contact sur les variétés fibrées en cercles audessus d'une surface, Comment. Math. Helv. 76 (2001) 218

[12] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309

[13] K Honda, On the classification of tight contact structures II, J. Differential Geom. 55 (2000) 83

[14] K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002) 435

[15] K Honda, W H Kazez, G Matić, Tight contact structures and taut foliations, Geom. Topol. 4 (2000) 219

[16] K Honda, W H Kazez, G Matić, Convex decomposition theory, Int. Math. Res. Not. (2002) 55

[17] K Honda, W H Kazez, G Matić, Tight contact structures on fibered hyperbolic 3–manifolds, J. Differential Geom. 64 (2003) 305

[18] W Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics 43, American Mathematical Society (1980)

[19] Y Kanda, The classification of tight contact structures on the 3–torus, Comm. Anal. Geom. 5 (1997) 413

[20] Y Kanda, On the Thurston–Bennequin invariant of Legendrian knots and nonexactness of Bennequin's inequality, Invent. Math. 133 (1998) 227

[21] S Makar-Limanov, Tight contact structures on solid tori, Trans. Amer. Math. Soc. 350 (1998) 1013

[22] J Martinet, Formes de contact sur les variétés de dimension 3, from: "Proceedings of Liverpool Singularities Symposium, II (1969/1970)", Springer (1971)

Cité par Sources :