Contact Ozsváth-Szabó invariants and Giroux torsion (Q2464791)

Giroux introduced the Giroux torsion invariant of a contact 3-manifold \((Y,\xi)\). For every natural number \(n\), Colin-Giroux-Honda proved that the 3-manifold \(Y\) carries at most finitely many isomorphism classes of tight contact structures with Giroux torsion bounded above by \(n\). In this paper the authors prove a vanishing theorem for the contact Ozsváth-Szabó invariants of \(OS_z\)-simple contact 3-manifolds having positive Giroux torsion. If a 3-manifold admits either a torus fibration over \(S^1\) or a Seifert fibration over an orientable base, then they show that there is an integer \(n\geq 0\) such that for every contact structure on the manifold with Giroux torsion \(>n\) the contact Ozsváth-Szabó invariant vanishes. Ghiggini found that the Brieskorn 3-spheres \(-\sum(2,3,12n+5)\) have strongly fillable contact structures which are not Stein fillable. The present authors use their vanishing results to show that the Brieskorn homology 3-spheres \(-\sum(2,3,6n+5)\), \(n\geq 2\), carry strongly fillable contact structure which are not Stein fillable. Using standard techniques from contact topology, they show that if a contact 3-manifold \((Y,\xi)\) has positive Giroux torsion then there is a Stein cobordism from \((Y,\xi)\) to a contact 3-manifold \((Y,\xi')\) such that \((Y,\xi)\) is obtained from \((Y,\xi')\) by a Lutz modification.