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

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.