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Holomorphic disks and genus bounds

2004; Mathematical Sciences Publishers; Volume: 8; Issue: 1 Linguagem: Inglês

10.2140/gt.2004.8.311

1465-3060

Peter Ozsváth, Zoltán Szabó,

Homotopy and Cohomology in Algebraic Topology

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.