Mapping class group relations, Stein fillings, and planar open book decompositions

Wand, A. (2012) Mapping class group relations, Stein fillings, and planar open book decompositions. Journal of Topology, 5(1), pp. 1-14. (doi: 10.1112/jtopol/jtr025)

Abstract

The aim of this paper is to use mapping class group relations to approach the ‘geography’ problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that, for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple obstruction to planarity, which we interpret in terms of the existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non-planar structures that cannot be shown to be non-planar by other existing methods.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Wand, Dr Andy
• Authors: Wand, A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Topology
• Publisher: Oxford University Press
• ISSN: 1753-8416
• ISSN (Online): 1753-8424