Geometry & Topology, Vol. 8 (2004)
Paper no. 8, pages 311--334.
Holomorphic disks and genus bounds
Peter Ozsvath and Zoltan Szabo
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.
Thurston norm, Dehn surgery, Seifert genus, Floer homology, contact structures
AMS subject classification.
Primary: 57R58, 53D40.
Secondary: 57M27, 57N10.
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
Department of Mathematics, Columbia University
New York, NY 10025,
Institute for Advanced Study, Princeton, New Jersey
Department of Mathematics, Princeton University
Princeton, New Jersey 08544, USA
Email: firstname.lastname@example.org, email@example.com
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