Abelian cycles in the homology of the Torelli group

Geometry Topology Seminar
Monday, February 14, 2022 - 2:00pm for 1 hour (actually 50 minutes)
Erik Lindell – University of Stockholm – lindell@math.su.sehttps://www.su.se/profiles/erli2531-1.395127
Daniel Minahan

The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.