## Algebraic & Geometric Topology

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

Alexandra Pettet

#### Abstract

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

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

#### Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 725-740.

Dates
Revised: 5 May 2005
Accepted: 21 June 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796428

Digital Object Identifier
doi:10.2140/agt.2005.5.725

Mathematical Reviews number (MathSciNet)
MR2153110

Zentralblatt MATH identifier
1085.20016

#### Citation

Pettet, Alexandra. The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$. Algebr. Geom. Topol. 5 (2005), no. 2, 725--740. doi:10.2140/agt.2005.5.725. https://projecteuclid.org/euclid.agt/1513796428

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