Calculating the second rational cohomology group of the Torelli group

TL;DR

This paper explains recent advances in computing the second rational cohomology group of the Torelli group, combining key results from Hain and Kupers--Randal-Williams.

Contribution

It provides an exposition of crucial results enabling the calculation of the second rational cohomology group of the Torelli group, including the Johnson homomorphism.

Findings

Hain's calculation of the cup product pairing image

Kupers--Randal-Williams's calculation of the maximal algebraic subrepresentation

Exposition of these results with prerequisite material

Abstract

Minahan and the author recently proved results that allow the calculation of the second rational cohomology group of the Torelli group. This builds on two key ingredients: Hain's calculation of the image of the cup product pairing on the first cohomology group, and Kupers--Randal-Williams's calculation of the maximal algebraic subrepresentation of the second cohomology group. This paper gives an exposition of both of these results, including prerequisite material about the Johnson homomorphism.