2009•arXiv (Cornell University)Open access

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

Takao Satoh

Abstract

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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Key concepts: Automorphism, Outer automorphism group, Inner automorphism, Group (periodic table), Alternating group, Mathematics, Cohomology, SO(8)