The sutured Floer homology polytope

Juhász, András

doi:10.2140/gt.2010.14.1303

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The sutured Floer homology polytope

András Juhász

Geometry & Topology 14 (2010) 1303–1354

DOI: 10.2140/gt.2010.14.1303

Abstract

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality and then define a polytope $P (M, γ)$ in $H^{2} (M, \partial M; ℝ)$ that is spanned by the ${Spin}^{c}$ –structures which support nonzero Floer homology groups. If $(M, γ) ⇝ (M^{'}, γ^{'})$ is a taut surface decomposition, then an affine map projects $P (M^{'}, γ^{'})$ onto a face of $P (M, γ)$ ; moreover, if $H_{2} (M) = 0$ , then every face of $P (M, γ)$ can be obtained in this way for some surface decomposition. We show that if $(M, γ)$ is reduced, horizontally prime and $H_{2} (M) = 0$ , then $P (M, γ)$ is maximal dimensional in $H^{2} (M, \partial M; ℝ)$ . This implies that if $rk (S F H (M, γ)) < 2^{k + 1}$ , then $(M, γ)$ has depth at most $2 k$ . Moreover, SFH acts as a complexity for balanced sutured manifolds. In particular, the rank of the top term of knot Floer homology bounds the topological complexity of the knot complement, in addition to simply detecting fibred knots.

Keywords

sutured manifold, Heegaard Floer homology, knot theory

Mathematical Subject Classification 2000

Primary: 57M27

Secondary: 57R58

References

Publication

Received: 16 February 2010

Revised: 2 April 2010

Accepted: 3 May 2010

Published: 31 May 2010

Proposed: David Gabai

Seconded: Peter Ozsváth, Tom Mrowka

Authors

András Juhász
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Wilberforce Road Cambridge CB3 0WB UK

Volume 14, issue 3 (2010)