Groupes de travail : Cohomology of Torelli groups

The mapping class groups of a (compact, orientable) surface is the group of isotopy classes of orientation-preserving self-diffeomorphisms of the surface. This group is an object of fundamental interest in low-dimensional topology, which also has important connections to several other areas, such as algebraic and hyperbolic geometry. The Torelli group of a surface is the subgroup of the mapping class group consisting of those elements which act trivially on the homology of the surface (this can be thought of as the "non-linear" part of the group). In comparison to mapping class groups, we know very little about Torelli groups, even when it comes to basic group theoretic properties, such as whether they are finitely presentable. A recent result of Minahan states that for surfaces of sufficiently large genus, the second rational cohomology of the Torelli group is finite dimensional, which rules out the simplest obstruction to finite presentability. The goal of this working group will be to understand this result of Minahan, and learn about cohomology of Torelli groups, and the methods used to study it, more generally. In this first talk, I will give a gentle introduction to mapping class groups and Torelli groups and survey some of what we know about their (rational) cohomology, without assuming any prior knowledge of the subject.