#### Algebraic and Geometric Topology 5 (2005),
paper no. 30, pages 725-740.

## The Johnson homomorphism and the second cohomology of IA_n

### Alexandra Pettet

**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^(i)
which act trivially on F_n/F_n^(i+1), the free rank n, degree i
nilpotent group. It is a conjecture of Andreadakis that K_n^(i) is
equal to the lower central series of IA_n; indeed K_n^(2) is known to
be the commutator subgroup of IA_n. We prove that the quotient group
K_n^(3)/IA_n^(3) is finite for all n and trivial for n=3. We also
compute the rank of K_n^(2)/K_n^(3).
**Keywords**.
Automorphisms of free groups, cohomology, Johnson homomorphism, descending central series

**AMS subject classification**.
Primary: 20F28, 20J06.
Secondary: 20F14.

**E-print:** `arXiv:math.GR/0501053`

**DOI:** 10.2140/agt.2005.5.725

Submitted: 13 January 2005.
(Revised: 5 May 2005.)
Accepted: 21 June 2005.
Published: 13 July 2005.

Notes on file formats
Alexandra Pettet

Department of Mathematics, University of Chicago

Chicago, IL 60637, USA

Email: alexandra@math.uchicago.edu

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