#### Volume 14, issue 4 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Stein fillable contact $3$–manifolds and positive open books of genus one

### Paolo Lisca

Algebraic & Geometric Topology 14 (2014) 2411–2430
##### 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

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.

##### Keywords
Stein fillings, contact structures, open books
Primary: 57R17
Secondary: 57R57