Dieter Kotschick

Abstract:   The mapping class group of a closed oriented surface consists of the isotopy classes of its self-diffeomorphisms. When the genus is at least two, this is a discrete group with very rich properties which appears naturally in several branches of algebra, geometry and topology.

We analyse mapping class groups by considering representations into them, which is equivalent to studying surface bundles. In the case when the base of a surface bundle is also a surface, the total space is a symplectic 4-manifold and methods of 4-dimensional symplectic geometry and gauge theory can be applied to obtain information about the mapping class groups. More generally, we consider the geometry of smooth Lefschetz fibrations and prove results about the bounded cohomology of mapping class groups. This also has purely algebraic consequences, like the fact that mapping class groups are not uniformly perfect.