First cohomologies and the Johnson homomorphisms of the automorphism group of a free group.
Let \(F_n\) be a free group of rank \(n\), let \(H\) be its Abelianisation, and let \(\Aut(F_n)\) be its automorphism group and \(\mathrm{Out}(F_n)\) be its outer automorphism group. Cohomology groups of \(\Aut(F_n)\) with coefficients in \(\mathbb Z\), \(\mathbb Q\), \(H\) and \(H^*\) (the dual of \(H\)) have been considered in the literature. This article considers the (first) cohomology group of \(\Aut(F_n)\) with coefficients in another \(\Aut(F_n)\)-module, \(V_L\), defined as follows. The kernel of the natural homomorphism \(\Aut(F_n)\to\Aut(H)\) induced from \(H\) is called the IA-automorphism group of \(F_n\) and is denoted \(IA_n\). Let \(V\) be the Abelianisation of \(IA_n\), let \(L\) be a principal ideal domain without 2-torsion, and set \(V_L=V\otimes_{\mathbb Z}L\). The main theorem is that for \(n\geq 5\) the first cohomology group \(H^1(\Aut(F_n),V_L)=L^{\oplus 2}\). Nielsen's presentation for \(\Aut(F_n)\) is central to the proof. Two applications are given (for \(n\geq 5\)). The first is that the first Johnson homomorphism \(\tau_1\colon IA_n\to V\) does not extend directly to \(\Aut(F_n)\) as a crossed homomorphism. The second is that \(H^1(\mathrm{Out}(F_n),V_L)=L\).
scientific article; zbMATH DE number 6155156
scientific article