Contractibility of fixed point sets of auter space

Abstract

We show that for every finite subgroup $G$ of $Aut(F_n)$, the fixed point subcomplex $X_n^G$ is contractible, where $F_n$ is the free group on $n$ letters and $X_n$ is the spine of ``auter space'' constructed by Hatcher and Vogtmann. In more categorical language, $X_n =\underline{E}Aut(F_n)$. This is useful because it allows one to compute the cohomology of normalizers or centralizers of finite subgroups of $Aut(F_n)$ based on their actions on fixed point subcomplexes. The techniques used to prove it are largely those of Krstic and Vogtmann, who in turn used techniques similar to Culler and Vogtmann.