#### Volume 5, issue 2 (2005)

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

### Alexandra Pettet

Algebraic & Geometric Topology 5 (2005) 725–740
 arXiv: math.GR/0501053
##### 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 ${K}_{n}^{\left(2\right)}∕{K}_{n}^{\left(3\right)}$.

##### Keywords
automorphisms of free groups, cohomology, Johnson homomorphism, descending central series
##### Mathematical Subject Classification 2000
Primary: 20F28, 20J06
Secondary: 20F14