Mapping class group relations, Stein fillings, and planar open book decompositions
arXiv:1006.2550 · 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 existence of certain configurations of curves in a factorization of the monodromy. We use these techniques to demonstrate examples of non-planar structures which cannot be shown non-planar by previously existing methods.
arXiv commentFinal version, includes minor stylistic revisions suggested by referee. Accepted for publication by the Journal of Topology
References in corpus (8)
- Géométrie de contact: de la dimension trois vers les dimensions supérieures
- An elementary approach to the mapping class group of a surface
- Weak Symplectic Fillings and Holomorphic Curves
- Planar open book decompositions and contact structures
- Planar open books and Floer homology
- A Presentation of the Mapping Class Groups
- On the topology of compact Stein surfaces
- On sections of genus two Lefschetz fibrations
Cited by in corpus (7)
- Calabi-Yau Caps, Uniruled Caps and Symplectic Fillings
- Contact surgery and supporting open books
- On symplectic fillings of spinal open book decompositions I: Geometric constructions
- Planar open books with four binding components
- Contact 3-manifolds, holomorphic curves and intersection theory
- Fillings of unit cotangent bundles of nonorientable surfaces
- On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification