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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.