Geometry & Topology

Holomorphic disks and genus bounds

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.

Article information

Source
Geom. Topol., Volume 8, Number 1 (2004), 311-334.

Dates
Revised: 12 February 2004
Accepted: 14 February 2004
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.gt/1513883369

Digital Object Identifier
doi:10.2140/gt.2004.8.311

Mathematical Reviews number (MathSciNet)
MR2023281

Zentralblatt MATH identifier
1056.57020

Citation

Ozsvath, Peter; Szabo, Zoltan. Holomorphic disks and genus bounds. Geom. Topol. 8 (2004), no. 1, 311--334. doi:10.2140/gt.2004.8.311. https://projecteuclid.org/euclid.gt/1513883369

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