Volume 8, issue 1 (2004)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Holomorphic disks and genus bounds

Peter Ozsvath and Zoltan Szabo

Geometry & Topology 8 (2004) 311–334

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 2000
Primary: 57R58, 53D40
Secondary: 57M27, 57N10
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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
Peter Ozsvath
Department of Mathematics
Columbia University
New York
New York 10025
USA
Institute for Advanced Study
Princeton
New Jersey 08540
USA
Zoltan Szabo
Department of Mathematics
Princeton University
Princeton
New Jersey 08544
USA