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.