Stein fillable contact $3$–manifolds and positive open books of genus one

Abstract

A $2$–dimensional open book $\left(S,h\right)$ determines a closed, oriented $3$–manifold $Y_{\left(S,h\right)}$ and a contact structure ${\xi }_{\left(S,h\right)}$ on $Y_{\left(S,h\right)}$. The contact structure ${\xi }_{\left(S,h\right)}$ is Stein fillable if $h$ is positive, ie $h$ can be written as a product of right-handed Dehn twists. Work of Wendl implies that when $S$ has genus zero the converse holds, that is

$${\xi }_{\left(S,h\right)}\text{ Stein fillable}\Rightarrow h\text{ positive}.$$

On the other hand, results by Wand [Phd thesis (2010)] and by Baker, Etnyre and Van Horn–Morris [J. Differential Geom. 90 (2012) 1-80] imply the existence of counterexamples to the above implication with $S$ of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the implication holds under the assumption that $S$ is a one-holed torus and $Y_{\left(S,h\right)}$ is a Heegaard Floer $L$–space.