### Actions of automorphism groups of free groups on homology spheres and acyclic manifolds

#### Version of 12 March 2008. Minor edits 15 October 2008. To appear in Commentarii Math Helv.

For \$n\ge 3\$, let \$SAut(F_n)\$ denote the unique subgroup of index two in the automorphism group of a free group of rank \$n\$.

The standard linear action of \$SL(n,Z)\$ on \$\R^n\$ induces non-trivial actions of \$SAut(F_n)\$ on \$\R^n\$ and on \$\S^{n-1}\$.
We prove that \$SAut(F_n)\$ admits no non-trivial actions by homeomorphisms on acyclic manifolds or spheres of smaller dimension.

Indeed, \$SAut(F_n)\$ cannot act non-trivially on any generalized \$\Z_2\$-homology sphere of dimension less than \$n-1\$,
nor on any \$\Z_2\$-acyclic \$\Z_2\$-homology manifold of dimension less than \$n\$.

It follows that \$SL(n,Z)\$ cannot act non-trivially on such spaces either.

When \$n\$ is even, we obtain similar results with \$\Z_3\$ coefficients.