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

TL;DR

This paper computes a specific twisted first cohomology group of the automorphism group of a free group, revealing its generators and implications for the extendability of the Johnson homomorphism.

Contribution

It explicitly calculates the cohomology group with novel generators derived from Magnus representations and shows limitations of the Johnson homomorphism extension.

Findings

The cohomology group is generated by two specific crossed homomorphisms.

The first Johnson homomorphism does not extend to the automorphism group for rank > 4.

Provides new insights into the structure of automorphism groups of free groups.

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.