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

Question

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?

Answer 1

In our paper here, Himes, Miller, Nariman, and myself prove that Hatcher-Vogtmann’s question has a negative answer, at least for $n=5$. It also probably has a negative answer for larger $n$, but our ignorance of the unstable cohomology of $Aut(F_n)$ prevents us from proving this via our techniques. I do not know any reasonable conjecture as to what the dualizing module is.