In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschetz fibrations. The simplest example is the lantern relation, already shown by the first author and Gurtas ('Lantern relations and rational blowdowns ', Proc. Amer. Math. Soc. 138 (2010) 1131-1142 to correspond to rational blowdown along a −4 sphere; here we give relations that extend that result to realize the 'generalized' rational blowdowns of Fintushel and Stern ('Rational blowdowns of smooth 4-manifolds', J. Differential Geom. 46 (1997) 181-235) and Park ('Seiberg-Witten invariants of generalised rational blow-downs ', Bull. Austral. Math. Soc. 56 (1997) 363-384) by monodromy substitution, as well as several of the families of rational blowdowns discovered by Stipsicz, Szabó, and Wahl ('Rational blowdowns and smoothings of surface singularities',

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.p; 1/ has a unique filling, and describe fillable and nonfillable tight contact structures on certain Seifert fibered spaces. 57R17; 53D35

Abstract. We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures -which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].

We compute the Ozsváth-Szabó contact invariants for all tight contact structures on the manifolds −Σ(2, 3, 6n − 1) using twisted coefficient and a previous computation by the first author and Ko Honda. This computation completes the classification of the tight contact structures in this family of 3-manifolds.