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

#### Martin R. Bridson and Karen Vogtmann

#### 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.