Erik Lindell: Stable cohomology of mapping class groups and Torelli groups

Abstract: The mapping class group of a compact, oriented surface is its group of self-diffeomorphisms, up to isotopy. A subgroup of fundamental interest is the Torelli group, consisting of those classes that act trivially on the homology of the surface. Until recently, very little was understood about the cohomology of these groups, but a few remarkable results in the last few years have led to the complete calculation of its stable cohomology, i.e., the cohomology in a range of degrees sufficiently small compared to the genus of the surface. A key ingredient in this calculation is the stable cohomology of the mapping class group, with certain twisted coefficients.

In this talk, I will give an overview of this recent progress, as well as parallell progress (partly due to myself) in the study of closely related groups in low-dimensional topology, such as automorphism groups of free groups and handlebody groups (the latter based on joint work with Soulié and Fernandes–Soulié).