A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group

Satoh, Takao

doi:10.1090/S0002-9947-2010-05227-4

A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group
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Trans. Amer. Math. Soc. 363 (2011), 1631-1664 Request permission

Abstract:

Let $F_n$ be a free group of rank $n$ and $F_n^N$ the quotient group of $F_n$ 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 $F_n$. In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map $F_n \rightarrow F_n^N$, we see that there appear only two irreducible components, the Morita obstruction $S^k H_{\mathbf {Q}}$ and the Schur-Weyl module of type $H_{\mathbf {Q}}^{[k-2, 1^2]}$, in the cokernel of the rational Johnson homomorphism $\tau _{k, \mathbf {Q}}’=\tau _k’ \otimes \mathrm {id}_{\mathbf {Q}}$ for $k \geq 5$ and $n \geq k+2$.

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