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

TL;DR

This paper proves that for a one-holed torus, Stein fillability of the associated contact structure is equivalent to the monodromy being a product of right-handed Dehn twists, under the condition that the manifold is a Heegaard Floer L-space.

Contribution

It establishes the converse of known results for genus one surfaces with one boundary component, linking Stein fillability to positive monodromy in this specific case.

Findings

Stein fillability is equivalent to positive monodromy for genus one, one-holed torus cases.

The result applies when the three-manifold is a Heegaard Floer L-space.

Provides a partial classification of Stein fillable contact structures in this setting.

Abstract

A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpose of this paper is to prove the converse statement under the assumption that S is a one-holed torus and Y(S,h) is a Heegaard Floer L-space.