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The classical formulation of Hilbert’s fifth problem asks whether topological groups that have the topological structure of a manifold, are necessarily Lie groups. This is indeed, the case, thanks to following theorem of Gleason and Montgomery-Zippin:
We have discussed the proof of this result, and of related results, in previous posts. There is however a generalisation of Hilbert’s fifth problem which remains open, namely the Hilbert-Smith conjecture, in which it is a space acted on by the group which has the manifold structure, rather than the group itself:
Conjecture 2 (Hilbert-Smith conjecture) Let be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold . Then is isomorphic to a Lie group.
Note that Conjecture 2 easily implies Theorem 1 as one can pass to the connected component of a locally Euclidean group (which is clearly locally compact), and then look at the action of on itself by left-multiplication.
The hypothesis that the action is faithful (i.e. each non-identity group element acts non-trivially on ) cannot be completely eliminated, as any group will have a trivial action on any space . The requirement that be locally compact is similarly necessary: consider for instance the diffeomorphism group of, say, the unit circle , which acts on but is infinite dimensional and is not locally compact (with, say, the uniform topology). Finally, the connectedness of is also important: the infinite torus (with the product topology) acts faithfully on the disconnected manifold by the action
The conjecture in full generality remains open. However, there are a number of partial results. For instance, it was observed by Montgomery and Zippin that the conjecture is true for transitive actions, by a modification of the argument used to establish Theorem 1. This special case of the Hilbert-Smith conjecture (or more precisely, a generalisation thereof in which “finite-dimensional manifold” was replaced by “locally connected locally compact finite-dimensional”) was used in Gromov’s proof of his famous theorem on groups of polynomial growth. I record the argument of Montgomery and Zippin below the fold.
Another partial result is the reduction of the Hilbert-Smith conjecture to the -adic case. Indeed, it is known that Conjecture 2 is equivalent to
The reduction to the -adic case follows from the structural theory of locally compact groups (specifically, the Gleason-Yamabe theorem discussed in previous posts) and some results of Newman that sharply restrict the ability of periodic actions on a manifold to be close to the identity. I record this argument (which appears for instance in this paper of Lee) below the fold also.