Mapping Class Groups of Del Pezzo Surfaces and Lefschetz Fibrations

This thesis comprises four papers on topics around the topological mapping class groups of del Pezzo and rational surfaces and the topology of 4-dimensional Lefschetz fibrations. Chapters 2 and 3 address the Nielsen realization problem for various finite subgroups of the mapping class groups of del Pezzo surfaces. Chapter 4, which is joint work with Carlos A. Serván, exhibits examples of Lefschetz fibrations with infinitely many homologically distinct symplectic sections. Finally, Chapter 5 studies the stabilizer of isotropic homology classes of minimal genus 0 in rational 4-manifolds under the action of the mapping class group.