Takao Satoh
Published 2009-01-06, updated 2012-05-18Version 3
We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.
The stable cohomology of automorphisms of free groups with coefficients in the homology representation
Stable rational homology of the IA-automorphism groups of free groups
On the structures of the Johnson cokernels of the basis-conjugating automorphism groups of free groups
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