Georgia Institute of Technology College of Sciences

Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial. Most of this beginner-friendly talk will be dedicated to proving the non-triviality of the first MMM class. Maybe as a side quest, we will also give a crash course on the geometric viewpoint of (co)homology and then apply this viewpoint to understand the constructions and the proofs.