Geometry & Topology

Holomorphic disks and genus bounds

Peter Ozsvath and Zoltan Szabo

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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

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

Received: 3 December 2003
Revised: 12 February 2004
Accepted: 14 February 2004
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R58: Floer homology 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 57M27: Invariants of knots and 3-manifolds 57N10: Topology of general 3-manifolds [See also 57Mxx]

Thurston norm Dehn surgery Seifert genus Floer homology contact structures


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

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