A note on Thurston-Winkelnkemper's construction of contact forms on 3-manifolds
In 1975 \textit{W. Thurston} and \textit{H. Winkelnkemper} [Proc. Am. Math. Soc. 52, 345-347 (1975; Zbl 0312.53028)] published an elegant proof that every oriented, closed, compact 3-manifold \(M\) admits a contact form \(\alpha\). The method exploits the existence of an open book decomposition for \(M\), with page a punctured surface \(F\) and spine \(S^1.\) By a careful combination of Dehn twists with the 'join up' construction of \(\alpha \) across the spine the author sharpens this result to show that the contact structure associated to the form alpha is symplectically fillable in the case when the monodromy is a composition of right-handed Dehn twists along mutually disjoint curves. As a corollary he also shows how to deform the structure obtained into a foliation having a Reeb component. Both results are nice illustrations of more general theorems.
scientific article; zbMATH DE number 1747453