Seifert fibered contact three-manifolds via surgery

Lisca, Paolo¹ ; Stipsicz, Andras I²

¹Dipartimento di Matematica, Università di Pisa, I-56127 Pisa, Italy
²Rényi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Reáltanoda utca 13–15, Hungary

Algebraic and Geometric Topology, Tome 4 (2004) no. 1, pp. 199-217

Résumé

Using contact surgery we define families of contact structures on certain Seifert fibered three–manifolds. We prove that all these contact structures are tight using contact Ozsváth–Szabó invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three–manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.

Détail

DOI : 10.2140/agt.2004.4.199

Keywords: Seifert fibered 3–manifolds, tight, fillable contact structures, Ozsváth–Szabó invariants

Bibliographie

[1] F Ding, H Geiges, Symplectic fillability of tight contact structures on torus bundles, Algebr. Geom. Topol. 1 (2001) 153

[2] F Ding, H Geiges, A Legendrian surgery presentation of contact 3–manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004) 583

[3] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact 3–manifolds, Turkish J. Math. 28 (2004) 41

[4] S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279

[5] Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990) 29

[6] J B Etnyre, K Honda, Tight contact structures with no symplectic fillings, Invent. Math. 148 (2002) 609

[7] E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: "Proceedings of the International Congress of Mathematicians, Vol II (Beijing, 2002)", Higher Ed. Press (2002) 405

[8] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. $(2)$ 148 (1998) 619

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

[10] P Lisca, Symplectic fillings and positive scalar curvature, Geom. Topol. 2 (1998) 103

[11] P Lisca, On fillable contact structures up to homotopy, Proc. Amer. Math. Soc. 129 (2001) 3437

[12] P Lisca, A I Stipsicz, An infinite family of tight, not semi-fillable contact three-manifolds, Geom. Topol. 7 (2003) 1055

[13] P Lisca, A I Stipsicz, Tight, not semi-fillable contact circle bundles, Math. Ann. 328 (2004) 285

[14] T Mrowka, P Ozsváth, B Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997) 685

[15] L I Nicolaescu, Finite energy Seiberg–Witten moduli spaces on 4–manifolds bounding Seifert fibrations, Comm. Anal. Geom. 8 (2000) 1027

[16] P Orlik, Seifert manifolds, Lecture Notes in Mathematics 291, Springer (1972)

[17] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[18] P Ozsváth, Z Szabó, Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. $(2)$ 159 (2004) 1159

[19] P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179

[20] P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39

[21] A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241