  We consider finite groups which admit a faithful, smooth action on an acyclic
manifold of dimension three, four or five (e.g. euclidean space). Our first
main result states that a finite group acting on an acyclic 3- or 4-manifold is
isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively.
The analogue remains open in dimension five (where it is not true for arbitrary
continuous actions, however). We prove that the only finite nonabelian simple
groups admitting a smooth action on an acyclic 5-manifold are the alternating
groups A_5 and A_6, and deduce from this a short list of finite groups, closely
related to the finite subgroups of SO(5), which are the candidates for
orientation-preserving actions on acyclic 5-manifolds.
