Monodromy substitutions and rational blowdowns

Abstract

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’, J. Topol. 1 (2008) 477–517).