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.

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Alexandra Pettet
Department of Mathematics, University of Chicago
Chicago, IL 60637, USA

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