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Let $M^{r}$ be a connected orientable manifold with the Euler characteristic$\chi(M)\not \equiv 0\operatorname{mod}6$. Denote by $\mathrm{SAut}(F_{n})$ theunique subgroup of index two in the automorphism group of a free group. Thenany group action of $\mathrm{SAut}(F_{n})$ (and thus the special linear group$\mathrm{SL}_{n}(\mathbb{Z})$) $(n\geq r+2$) on $M^{r}$ by homeomorphisms istrivial. This confirms a conjecture related to Zimmer's program for thesemanifolds.

Euler characteristics and actions of automorphism groups of free groups