Holomorphic disks and genus bounds

Ozsvath, Peter¹ ; Szabo, Zoltan²

¹ Department of Mathematics, Columbia University, New York, New York 10025, USA, Institute for Advanced Study, Princeton, New Jersey 08540, USA
² Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA

Geometry & topology, Tome 8 (2004) no. 1, pp. 311-334.

Résumé

We prove that, like the Seiberg–Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to new proofs of certain results previously obtained using Seiberg–Witten monopole Floer homology (in collaboration with Kronheimer and Mrowka). It also leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof shows that the canonical element of Heegaard Floer homology associated to a weakly symplectically fillable contact structure is non-trivial. In particular, for certain three-manifolds, Heegaard Floer homology gives obstructions to the existence of taut foliations.

Détail

DOI : 10.2140/gt.2004.8.311

Keywords: Thurston norm, Dehn surgery, Seifert genus, Floer homology, contact structures

Bibliographie

[1] S Akbulut, B Ozbagci, Lefschetz fibrations on compact Stein surfaces, Geom. Topol. 5 (2001) 319

[2] S K Donaldson, Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999) 205

[3] Y Eliashberg, A few remarks about symplectic filling, Geom. Topol. 8 (2004) 277

[4] Y Eliashberg, M Gromov, Convex symplectic manifolds, from: "Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989)", Proc. Sympos. Pure Math. 52, Amer. Math. Soc. (1991) 135

[5] Y M Eliashberg, W P Thurston, Confoliations, University Lecture Series 13, American Mathematical Society (1998)

[6] J B Etnyre, On symplectic fillings, Algebr. Geom. Topol. 4 (2004) 73

[7] J B Etnyre, K Honda, On symplectic cobordisms, Math. Ann. 323 (2002) 31

[8] D Gabai, Foliations and the topology of 3–manifolds, J. Differential Geom. 18 (1983) 445

[9] D Gabai, Foliations and the topology of 3–manifolds III, J. Differential Geom. 26 (1987) 479

[10] D T Gay, Explicit concave fillings of contact three-manifolds, Math. Proc. Cambridge Philos. Soc. 133 (2002) 431

[11] 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

[12] R E Gompf, A I Stipsicz, 4–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999)

[13] C M Gordon, Some aspects of classical knot theory, from: "Knot theory (Proc. Sem., Plans-sur–Bex, 1977)", Lecture Notes in Math. 685, Springer (1978) 1

[14] C M Gordon, J Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371

[15] L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton University Press (1987)

[16] P B Kronheimer, T S Mrowka, Floer homology for Seiberg–Witten Monopoles, in preparation

[17] P B Kronheimer, T S Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997) 209

[18] P B Kronheimer, T S Mrowka, Scalar curvature and the Thurston norm, Math. Res. Lett. 4 (1997) 931

[19] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. $(2)$ 165 (2007) 457

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

[21] P Lisca, A I Stipsicz, Seifert fibered contact three-manifolds via surgery, Algebr. Geom. Topol. 4 (2004) 199

[22] P Lisca, G Matić, Tight contact structures and Seiberg–Witten invariants, Invent. Math. 129 (1997) 509

[23] J Milnor, A unique decomposition theorem for 3–manifolds, Amer. J. Math. 84 (1962) 1

[24] A Némethi, On the Ozsváth–Szabó invariant of negative definite plumbed 3–manifolds, Geom. Topol. 9 (2005) 991

[25] H Ohta, K Ono, Simple singularities and topology of symplectically filling 4–manifold, Comment. Math. Helv. 74 (1999) 575

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

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

[28] P Ozsváth, Z Szabó, Holomorphic triangles and invariants for smooth four-manifolds, Adv. Math. 202 (2006) 326

[29] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281

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

[31] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

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

[33] P Ozsváth, Z Szabó, Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003) 225

[34] P Ozsváth, Z Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004) 1

[35] J Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)

[36] C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809

[37] C H Taubes, More constraints on symplectic forms from Seiberg–Witten invariants, Math. Res. Lett. 2 (1995) 9

[38] C H Taubes, Seiberg Witten and Gromov invariants for symplectic 4–manifolds, First International Press Lecture Series 2, International Press (2000)

[39] W P Thurston, A norm for the homology of 3–manifolds, Mem. Amer. Math. Soc. 59 (1986)

[40] V Turaev, Torsion invariants of $\mathrm{Spin}^c$–structures on 3–manifolds, Math. Res. Lett. 4 (1997) 679

[41] E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769