The Johnson homomorphism and the second cohomology of IA_n

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).