Twisted second homology groups of the automorphism group of a free group. (Q995616)

In this paper the author computes the second homology groups of the automorphism group of a free group with coefficients in the Abelianized free group and its dual group. Let \(H\) be the Abelianization of the free group \(F_n\) (i.e. \(H=F_n /[F_n,F_n]\)) and \(H^*=\Hom_\mathbb{Z}(H,\mathbb{Z})\) its dual. Let \(L\) be the subring \(\mathbb{Z}[\tfrac 12]\) of \(\mathbb{Q}\) obtained from \(\mathbb{Z}\) by attaching \(\tfrac 12\). The ring \(L\) is a principal ideal domain in which the element 2 is invertible. Let \(H_L\) denote the \(L\)-module \(H\otimes_\mathbb{Z} L\) and \(H^*_L\) its dual. The main theorem of the paper is: For \(n\geq 6\), \(H_2(\Aut F_n,H_L)=0\), \(H_2(\Aut F_n,H^*_L)=0\). -- The proof is by using combinatorial group theory based on Gersten's presentation of \(\Aut^+F_n\) (\(\Aut^+F_n\) the special automorphism group of \(F_n\)). At the end the author states two problems concerning the second homology and cohomology groups of \(\Aut F_n\) with coefficients in a certain ring.