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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

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 fillableh 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.

Stein fillings, contact structures, open books
Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57R57
Received: 10 April 2013
Revised: 5 January 2014
Accepted: 7 January 2014
Published: 28 August 2014
Paolo Lisca
Dipartimento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5
56121 Pisa