First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group

Details First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group

2009-01-06

arxiv.org/abs/0901.0589v2

Mathematics

Algebraic Topology

26 pages

Scientific paper

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.

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