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

Lisca, Paolo

doi:10.2140/agt.2014.14.2411

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

Paolo Lisca

Algebraic & Geometric Topology 14 (2014) 2411–2430

DOI: 10.2140/agt.2014.14.2411

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

ξ_{(S, h)} Stein fillable \Rightarrow h 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 $Y_{(S, h)}$ is a Heegaard Floer $L$ –space.

Keywords

Stein fillings, contact structures, open books

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57R57

References

Publication

Received: 10 April 2013

Revised: 5 January 2014

Accepted: 7 January 2014

Published: 28 August 2014

Authors

Paolo Lisca
Dipartimento di Matematica Università di Pisa Largo Bruno Pontecorvo 5 56121 Pisa Italy

Volume 14, issue 4 (2014)