The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$

Abstract

Let $F_n$ be the free group on $n$ generators. Define ${IA}_n$ to be group of automorphisms of $F_n$ that act trivially on first homology. The Johnson homomorphism in this setting is a map from ${IA}_n$ to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of ${IA}_n$.

A descending central series of ${IA}_n$ is given by the subgroups $K_n^{\left(i\right)}$ which act trivially on $F_n∕F_n^{\left(i+1\right)}$, the free rank $n$, degree $i$ nilpotent group. It is a conjecture of Andreadakis that $K_n^{\left(i\right)}$ is equal to the lower central series of ${IA}_n$; indeed $K_n^{\left(2\right)}$ is known to be the commutator subgroup of ${IA}_n$. We prove that the quotient group $K_n^{\left(3\right)}∕{IA}_n^{\left(3\right)}$ is finite for all $n$ and trivial for $n=3$. We also compute the rank of $K_n^{\left(2\right)}∕K_n^{\left(3\right)}$.