# Mapping class groups

Graduate topics course, *Harvard University*, 2017

Here are hand-written notes from the course.

This was a course on mapping class groups of surfaces. Topics included:

- Algebraic structure of mapping class groups
- finite generation
- presentation
- abelianization

- Surface bundles
- Over the circle: multiple fiberings, Thurston norm
- Over surfaces: signature, Atiyah–Kodaira examples, surface subgroups of Modg
- Characteristic classes: Miller–Morita–Mumford classes

- Mumford conjecture.
- Harer’s homological stability theorem
- Madsen–Weiss theorem
- Homotopy type of diffeomorphism groups, Earle–Eells theorem

- Lifting problems for Mod_g
- Morita’s nonlifting theorem, flat connections on surface bundles
- Lifting problems for surface braid groups