Mapping class groups and fiber bundles (Lecture 2)

We welcome the entire department to this Frontiers series! The mapping class group Mod(M) of a smooth manifold M is the group of diffeomorphisms of M, modulo isotopy. Mapping class groups play an important role in geometric topology, especially in low dimensions, and they have connections to geometric group theory, dynamics, homotopy theory, algebraic geometry, and more. In these lectures we will give a broad overview of the theory of mapping class groups and then discuss two problems about these groups that relate to the study of fiber bundles. The first is the Nielsen realization problem, which asks when a subgroup of Mod(M) can be lifted to the diffeomorphism group Diff(M). The second is the “monodromy arithmeticity problem” of Griffiths-Schmid, which arises in the study of families of Riemann surfaces.