Geometry & Topology 8 (2004) 311–334

DOI: 10.2140/gt.2004.8.311

arXiv: math.GT/0311496

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

Mathematical Subject Classification

Primary: 53D40, 57R58

Secondary: 57M27, 57N10

References

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Publication

Received: 3 December 2003
Revised: 12 February 2004
Accepted: 14 February 2004
Published: 14 February 2004
Proposed: Robion Kirby
Seconded: John Morgan, Ronald Stern

Authors