On the top homology of the Torelli group and the Johnson kernel

Mapping class groups of oriented surfaces are closely related to moduli spaces of complex curves, and to topology of 3-manifolds. The “non-linear part” of the mapping class group is the Torelli subgroup $\mathcal{I}_g$, consisting of all mapping classes acting trivially on the first homology of the surface. From a topological point of view, this subgroup is interesting because of its connection to homology 3-spheres. The Torelli group also arises in algebraic geometry as the fundamental group of the Torelli space, the moduli space of smooth complex curves with homology framings. The first homology group ${\rm H}_1(\mathcal{I}_g, \mathbb{Z})$ was described by Johnson in the 1980's. However, none of the other non-zero homology groups has been computed explicitly. In this talk, we discuss the problem of computing the homology of the Torelli group and the Johnson kernel $\mathcal{K}_g$, which is the most well-studied subgroup of $\mathcal{I}_g$. We give a complete description of the Torelli group top homology group in genus 3. Also, we describe the subgroup of the Johnson kernel top homology group, generated by abelian cycles determined by Dehn twists about disjoint separating curves. The main approach is the study of the action of $\mathcal{I}_g$ and $\mathcal{K}_g$ on the complex of cycles, introduced by Bestvina, Bux and Margalit in 2007.