Geometry & Topology, Vol. 8 (2004) Paper no. 8, pages 311--334.

Holomorphic disks and genus bounds

Peter Ozsvath and Zoltan Szabo

Abstract. 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.

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

AMS subject classification. Primary: 57R58, 53D40. Secondary: 57M27, 57N10.

DOI: 10.2140/gt.2004.8.311

E-print: arXiv:math.GT/0311496

Submitted to GT on 3 December 2003. (Revised 12 February 2004.) Paper accepted 14 February 2004. Paper published 14 February 2004.

Notes on file formats

Peter Ozsvath
Department of Mathematics, Columbia University
New York, NY 10025, USA
and
Institute for Advanced Study, Princeton, New Jersey 08540, USA

Zoltan Szabo
Department of Mathematics, Princeton University
Princeton, New Jersey 08544, USA

Email: petero@math.columbia.edu, szabo@math.princeton.edu

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