The Hilbert-Smith Conjecture

Abstract

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (``Finite-to-one mappings of manifolds'', Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a p-adic group on compact connected n-manifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities. It is well known that if a locally compact group acts effectively on a connected n-manifold M and G is not a Lie group, then there is a subgroup H of G isomorphic to a p-adic group A_p which acts effectively on M. It can be shown that A_p can not act effectively on an n-manifold and, hence, The Hilbert Smith Conjecture is true. The existence of a non empty fixed point set adds some complexity to the proof. In this paper, it is shown that A_p can not act freely on a compact connected n-manifold. The basic ideas for the general case are more clearly seen in this case. The general proof will be given in another paper.