The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group

Satoh, Takao

doi:10.1090/S0002-9947-08-04767-3

The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group
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Trans. Amer. Math. Soc. 361 (2009), 2085-2107 Request permission

Abstract:

In this paper, we determine the cokernel of the $k$-th Johnson homomorphisms of the automorphism group of a free metabelian group for $k \geq 2$ and $n \geq 4$. As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.

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