A REDUCTION OF THE TARGET OF THE JOHNSON HOMOMORPHISMS OF THE AUTOMORPHISM GROUP OF A FREE GROUP

作者：Satoh Takao^*

来源：Transactions of the American Mathematical Society, 2011, 363(3): 1631-1664.

• Automorphism group of a free group
• IA-automorphism group
• Johnson homomorphisms
• Chen's integration on free groups

Let F(n) be a free group of rank n and F(n)(N) the quotient group of F by a subgroup [Gamma(n) (3), Gamma(n) (3)][[Gamma(n) (2), Gamma(n) (2)], Gamma(n),(2)], where Gamma(n)(k) denotes the k-th subgroup of the lower central series of the free group F(n). In this paper, we determine the group structure of the graded quotients of the lower central series of the group F(n)(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 F. In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map F(n) -> F(n)(N), we see that there appear only two irreducible components, the Morita obstruction S(k)H(Q) and the Schur-Weyl module of type H(Q)([k-2,12]) , in the cokernel of the rational Johnson homomorphism tau'(k),(Q) = tau'(k) circle times id(Q) for k >= 5 and is n >= k + 2.

• 出版日期2011-3