Algebraic and Geometric Topology 5 (2005),
paper no. 30, pages 725-740.
The Johnson homomorphism and the second cohomology of IA_n
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).
Automorphisms of free groups, cohomology, Johnson homomorphism, descending central series
AMS subject classification.
Primary: 20F28, 20J06.
Submitted: 13 January 2005.
(Revised: 5 May 2005.)
Accepted: 21 June 2005.
Published: 13 July 2005.
Notes on file formats
Department of Mathematics, University of Chicago
Chicago, IL 60637, USA
AGT home page
These pages are not updated anymore.
They reflect the state of
For the current production of this journal, please refer to