Monoids in the mapping class group

In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.Comment: 36 pages, 18 figure

Monodromy Substitutions and Rational Blowdowns

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.Comment: 28 pages, many figures. v2: minor edits; this version accepted for publication in Journal of Topolog