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Torsion and open book decompositions. (English) Zbl 1213.57028
The paper studies open book decompositions of contact manifolds. The binding of the open book is a transverse link in the contact manifold, and the authors use link invariants coming from Heegaard Floer theory to show that the complement of the binding has no Giroux torsion. They also prove the stronger result that no finite cover of the complement of the binding contains positive Giroux torsion. The key tool in their proof is the invariant for oriented Legendrian and transverse links studied in A. I. Stipsicz and V. Vértesi [Pac. J. Math. 239, No. 1, 157–177 (2009; Zbl 1149.57031)].
Viewing contact manifolds with convex boundary as sutured manifolds, Honda, Kazez, and Matić defined an invariant \(c(Y, \xi)\) in the sutured Floer homology of the oppositely-oriented manifold, cf. K. Honda, W. H. Kazez, and G. Matić [Invent. Math. 176, No. 3, 637–676 (2009; Zbl 1171.57031)]. If \((Y, \xi)\) has Giroux torsion, then this contact invariant \(c(Y,\xi)\) vanishes, cf. P. Ghiggini, K. Honda, and J. Van Horn-Morris [The vanishing of the contact invariant in the presence of torsion. arXiv:0706.1602]. Stipsicz and Vertesi map the contact invariant associated to the sutured complement of a Legendrian knot \(L\) to the Legendrian knot invariant \(\hat{\mathcal{L}}(L)\) defined in P. Lisca, P. Ozsváth, A. I. Stipsicz, and Z. Szabó [J. Eur. Math. Soc. (JEMS) 11, No. 6, 1307–1363 (2009; Zbl 1232.57017)]. This gives rise to an invariant of transverse knots, denoted here by \(\bar{c}(L)\), which vanishes when the complement of \(L\) has Giroux torsion. In this paper, the authors show that for an open book \((B, \pi)\), \(\bar{c}(B)\neq 0\).
The proof itself relies on two lemmas. First, a negative basic slice is glued to the complement of a Legendrian approximation of the binding, and Colin’s gluing theorem is used to show that the resulting contact manifold is universally tight, V. Colin [Bull. Soc. Math. Fr. 127, No. 1, 43–69 (1999; Zbl 0930.53053)]. The second lemma establishes the non-vanishing of the contact invariant \(\bar{c}(B)\), which proves that the complement of the binding has no Giroux torsion. In the proof of the second lemma, the authors cut the given open book along a well-groomed surface. The resulting tight contact manifold embeds in a closed Stein fillable contact manifold, and they apply a theorem of K. Honda, W. H. Kazez and G. Matić [loc. cit.] to complete the argument.

MSC:
57R17 Symplectic and contact topology in high or arbitrary dimension
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R58 Floer homology
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