A Reduction of the Target of the Johnson Homomorphisms of the Automorphism Group of a Free Group

Let Fn be a free group of rank n and F n the quotient group of Fn by the subgroup [Γn(3),Γn(3)][[Γn(2),Γn(2)],Γn(2)] where Γn(k) denotes the k-th subgroup of the lower central series of the free group Fn. In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n by using a generalized Chen’s integration in free groups. Then we apply it to the study of the Johnson homomorphisms of the automorphism group of Fn. In particular, after taking a reduction of the target of the Johnson homomorphism induced from a quotient map Fn → F n , we see that there appear only two irreducible component, the Morita obstruction SHQ and the Schur-Weyl module of type H [k−2,12] Q , in the cokernel of the rational Johnson homomorphism τ ′ k,Q = τ ′ k ⊗ idQ for k ≥ 5 and n ≥ k + 2.