Planar open books, monodromy factorizations and symplectic fillings

Abstract

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl’s theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on $L\left(p,1\right)$ has a unique filling, and describe fillable and nonfillable tight contact structures on certain Seifert fibered spaces.