# Dualizing module for $\operatorname{Aut}(F_n)$

In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $$FC_n$$ called the complex of free factors'' of the free group $$F_n$$. They proved that $$FC_n$$ is homotopy equivalent to a wedge of $$(n-2)$$-spheres. They call the top reduced homology group of this complex the Steinberg module of $$\operatorname{Aut}(F_n)$$ and ask if it is a rational dualizing module for $$\operatorname{Aut}(F_n)$$. That is, they ask if: $$H^i(\operatorname{Aut}(F_n);\mathbb Q) \cong H_{2n-2-i}(\operatorname{Aut}(F_n);\tilde H_{n-2}(FC_n;\mathbb Q)).$$ Has there been any progress on this question?