## Algebraic & Geometric Topology

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

Paolo Lisca

#### Abstract

A $2$–dimensional open book $(S,h)$ determines a closed, oriented $3$–manifold $Y(S,h)$ and a contact structure $ξ(S,h)$ on $Y(S,h)$. The contact structure $ξ(S,h)$ 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 $Y(S,h)$ is a Heegaard Floer $L$–space.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2411-2430.

Dates
Revised: 5 January 2014
Accepted: 7 January 2014
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715967

Digital Object Identifier
doi:10.2140/agt.2014.14.2411

Mathematical Reviews number (MathSciNet)
MR3331617

Zentralblatt MATH identifier
1300.57027

#### Citation

Lisca, Paolo. Stein fillable contact $3$–manifolds and positive open books of genus one. Algebr. Geom. Topol. 14 (2014), no. 4, 2411--2430. doi:10.2140/agt.2014.14.2411. https://projecteuclid.org/euclid.agt/1513715967

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