Floer homology and surface decompositions

Juhász, András

doi:10.2140/gt.2008.12.299

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Floer homology and surface decompositions

András Juhász

Geometry & Topology 12 (2008) 299–350

DOI: 10.2140/gt.2008.12.299

Abstract

Sutured Floer homology, denoted by $S F H$ , is an invariant of balanced sutured manifolds previously defined by the author. In this paper we give a formula that shows how this invariant changes under surface decompositions. In particular, if $(M, γ) ⇝ (M^{'}, γ^{'})$ is a sutured manifold decomposition then $S F H (M^{'}, γ^{'})$ is a direct summand of $S F H (M, γ)$ . To prove the decomposition formula we give an algorithm that computes $S F H (M, γ)$ from a balanced diagram defining $(M, γ)$ that generalizes the algorithm of Sarkar and Wang.

As a corollary we obtain that if $(M, γ)$ is taut then $S F H (M, γ) \neq 0$ . Other applications include simple proofs of a result of Ozsváth and Szabó that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. Our proofs do not make use of any contact geometry.

Moreover, using these methods we show that if $K$ is a genus $g$ knot in a rational homology $3$ –sphere $Y$ whose Alexander polynomial has leading coefficient $a_{g} \neq 0$ and if $rk \hat{H F K} (Y, K, g) < 4$ then $Y ∖ N (K)$ admits a depth $\leq 2$ taut foliation transversal to $\partial N (K)$ .

Keywords

sutured manifold, Floer homology, surface decomposition

Mathematical Subject Classification 2000

Primary: 57M27, 57R58

References

Publication

Received: 13 November 2006

Accepted: 24 November 2007

Published: 12 March 2008

Proposed: Peter Ozsváth

Seconded: Ron Fintushel, Tom Mrowka

Authors

András Juhász
Department of Mathematics Princeton University Princeton NJ 08544 USA

Volume 12, issue 1 (2008)