1 A Proof Of The Hilbert - Smith Conjecture by Louis F . McAuley Dedicated to the memory of Deane Montgomery

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.