The sutured Floer homology polytope

Juhász, András¹

¹ Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

Geometry & topology, Tome 14 (2010) no. 3, pp. 1303-1354.

Résumé

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 $P\left(M,\gamma \right)$ in $H^2\left(M,\partial M;ℝ\right)$ that is spanned by the ${Spin}^c$–structures which support nonzero Floer homology groups. If $\left(M,\gamma \right)⇝\left(M^{\prime },{\gamma }^{\prime }\right)$ is a taut surface decomposition, then an affine map projects $P\left(M^{\prime },{\gamma }^{\prime }\right)$ onto a face of $P\left(M,\gamma \right)$; moreover, if $H_2\left(M\right)=0$, then every face of $P\left(M,\gamma \right)$ can be obtained in this way for some surface decomposition. We show that if $\left(M,\gamma \right)$ is reduced, horizontally prime and $H_2\left(M\right)=0$, then $P\left(M,\gamma \right)$ is maximal dimensional in $H^2\left(M,\partial M;ℝ\right)$. This implies that if $rk\left(SFH\left(M,\gamma \right)\right)<2^{k+1}$, then $\left(M,\gamma \right)$ has depth at most $2k$. 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.

Détail

DOI : 10.2140/gt.2010.14.1303

Keywords: sutured manifold, Heegaard Floer homology, knot theory

Bibliographie

[1] D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173

[2] Y M Eliashberg, W P Thurston, Confoliations, Univ. Lecture Ser. 13, Amer. Math. Soc. (1998)

[3] S Friedl, A Juhász, J Rasmussen, The decategorification of sutured Floer homology

[4] D Gabai, Foliations and the topology of $3$–manifolds, J. Differential Geom. 18 (1983) 445

[5] D Gabai, Foliations and genera of links, Topology 23 (1984) 381

[6] D Gabai, Detecting fibred links in $S^3$, Comment. Math. Helv. 61 (1986) 519

[7] D Gabai, Foliations and the topology of $3$–manifolds. II, J. Differential Geom. 26 (1987) 461

[8] D Gabai, Foliations and the topology of $3$–manifolds. III, J. Differential Geom. 26 (1987) 479

[9] P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151

[10] M Hedden, A Juhász, S Sarkar, On sutured Floer homology and the equivalence of Seifert surfaces

[11] K Honda, W H Kazez, G Matić, Convex decomposition theory, Int. Math. Res. Not. (2002) 55

[12] W H Jaco, P B Shalen, Seifert fibered spaces in $3$–manifolds, Mem. Amer. Math. Soc. 21 (1979)

[13] A Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006) 1429

[14] A Juhász, Floer homology and surface decompositions, Geom. Topol. 12 (2008) 299

[15] Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577

[16] Y Ni, Erratum: Knot Floer homology detects fibred knots \rm\codarefhttp://www.ams.org/mathscinet-getitem?mr=2357503[MR2357503], Invent. Math. 177 (2009) 235

[17] P Ozsváth, Z Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004) 311

[18] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58

[19] P Ozsváth, Z Szabó, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. $(2)$ 159 (2004) 1027

[20] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281

[21] P Ozsváth, Z Szabó, Holomorphic disks, link invariants and the multi-variable Alexander polynomial, Algebr. Geom. Topol. 8 (2008) 615

[22] P Ozsváth, Z Szabó, Link Floer homology and the Thurston norm, J. Amer. Math. Soc. 21 (2008) 671

[23] J A Rasmussen, Floer homology and knot complements, PhD Thesis, Harvard University, ProQuest LLC, Ann Arbor, MI (2003)

[24] M Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989) 557

[25] W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 59 (1986) 99