Abstract. 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.
It is a consequence of the classical Jordan bound for finite subgroups of\ud
linear groups that in each dimension n there are only finitely many finite simple groups\ud
which admit a faithful, linear action on the n-sphere. In the present paper we prove\ud
an analogue for smooth actions on arbitrary homology n-spheres: in each dimension\ud
n there are only finitely many finite simple groups which admit a faithful, smooth\ud
action on some homology sphere of dimension n, and in particular on the n-sphere.\ud
We discuss also the finite simple groups which admit an action on a homology sphere\ud
of dimension 3, 4 or 5
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