Contact Ozsváth–Szabó invariants and Giroux torsion

Abstract

In this paper we prove a vanishing theorem for the contact Ozsváth–Szabó invariants of certain contact 3–manifolds having positive Giroux torsion. We use this result to establish similar vanishing results for contact structures with underlying 3–manifolds admitting either a torus fibration over $S^1$ or a Seifert fibration over an orientable base. We also show – using standard techniques from contact topology – that if a contact 3–manifold $\left(Y,\xi \right)$ has positive Giroux torsion then there exists a Stein cobordism from $\left(Y,\xi \right)$ to a contact 3–manifold $\left(Y,{\xi }^{\prime }\right)$ such that $\left(Y,\xi \right)$ is obtained from $\left(Y,{\xi }^{\prime }\right)$ by a Lutz modification.