A note on Thurston-Winkelnkemper's construction of contact forms on 3-manifolds
type: Item
instance of: scholarly article
title: A note on Thurston-Winkelnkemper's construction of contact forms on 3-manifolds
zbMATH Open document ID: 1018.57012
author: Atsuhide Mori
published in: Osaka Journal of Mathematics
publication date: 2003-09-02T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1747453
zbMATH Keywords: 3-manifold
zbMATH Keywords: contact form
zbMATH Keywords: symplectically fillable
zbMATH Keywords: foliation
zbMATH Keywords: Reeb
reviewed by: Charles B. Thomas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1747453