Algebraic & Geometric Topology

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

Paolo Lisca

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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 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.

Article information

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

Dates
Received: 10 April 2013
Revised: 5 January 2014
Accepted: 7 January 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
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

Subjects
Primary: 57R17: Symplectic and contact topology
Secondary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]

Keywords
Stein fillings contact structures open books

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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