Lantern relations and rational blowdowns

TL;DR

This paper explores how the lantern relation in mapping class groups can be used to understand rational blowdowns in 4-manifolds, linking algebraic relations to topological modifications of Lefschetz fibrations.

Contribution

It establishes a novel connection between lantern relations and rational blowdowns, providing explicit examples of how relator modifications induce topological changes.

Findings

Relator modifications correspond to rational blowdowns in Lefschetz fibrations.

Examples show blowups are homeomorphic but not diffeomorphic to original fibrations.

The approach offers new insights into 4-manifold topology via mapping class group relations.

Abstract

We discuss a connection between the lantern relation in mapping class groups and the rational blowing down process for 4-manifolds. More precisely, if we change a positive relator in Dehn twist generators of the mapping class group by using a lantern relation, the corresponding Lefschetz fibration changes into its rational blowdown along a copy of the configuration C_2. We exhibit examples of such rational blowdowns of Lefschetz fibrations whose blowup is homeomorphic but not diffeomorphic to the original fibration.