Title: Obstructions to planarity of contact 3-manifolds,
Abstract: We prove that if a contact 3-manifold admits an open book decomposition of
genus 0, a certain intersection pattern cannot appear in the homology of any of
its symplectic fillings, and morever, fillings cannot contain certain
symplectic surfaces. Applying these obstructions to canonical contact
structures on links of normal surface singularities, we show that links of
isolated singularities of surfaces in the complex 3-space are planar only in
the case of $A_n$-singularities, and in general characterize completely planar
links of normal surface singularities (in terms of their resolution graphs). We
also establish non-planarity of tight contact structures on certain small
Seifert fibered L-spaces and of contact structures compatible with open books
given by a boundary multi-twist on a page of positive genus. Additionally, we
prove that every finitely presented group is the fundamental group of a
Leschetz fibration with planar fibers.