Monodromy Substitutions and Rational Blowdowns

TL;DR

This paper develops new relations in mapping class groups that correspond to rational blowdowns in 4-manifolds, extending known results and enabling geometric transformations via monodromy substitutions in Lefschetz fibrations.

Contribution

It introduces new families of relations in mapping class groups that realize generalized rational blowdowns through monodromy substitutions.

Findings

Relations extend lantern relation to generalized blowdowns

Realizes Fintushel-Stern and Park's rational blowdowns

Connects mapping class group relations with 4-manifold topology

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 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-Stern and Park by monodromy subsitution, as well as several of the families of rational blowdowns discovered by Stipsicz-Szab\'o-Wahl.