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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:

Theorem 1 (Hilbert’s fifth problem) Let {G} be a topological group which is locally Euclidean. Then {G} is isomorphic to a Lie group.

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 {G} be a locally compact topological group which acts continuously and faithfully (or effectively) on a connected finite-dimensional manifold {X}. Then {G} is isomorphic to a Lie group.

Note that Conjecture 2 easily implies Theorem 1 as one can pass to the connected component {G^\circ} of a locally Euclidean group (which is clearly locally compact), and then look at the action of {G^\circ} on itself by left-multiplication.

The hypothesis that the action is faithful (i.e. each non-identity group element {g \in G \backslash \{\hbox{id}\}} acts non-trivially on {X}) cannot be completely eliminated, as any group {G} will have a trivial action on any space {X}. The requirement that {G} be locally compact is similarly necessary: consider for instance the diffeomorphism group {\hbox{Diff}(S^1)} of, say, the unit circle {S^1}, which acts on {S^1} but is infinite dimensional and is not locally compact (with, say, the uniform topology). Finally, the connectedness of {X} is also important: the infinite torus {G = ({\bf R}/{\bf Z})^{\bf N}} (with the product topology) acts faithfully on the disconnected manifold {X := {\bf R}/{\bf Z} \times {\bf N}} by the action

\displaystyle  (g_n)_{n \in {\bf N}} (\theta, m) := (\theta + g_m, m).

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 {p}-adic case. Indeed, it is known that Conjecture 2 is equivalent to

Conjecture 3 (Hilbert-Smith conjecture for {p}-adic actions) It is not possible for a {p}-adic group {{\bf Z}_p} to act continuously and effectively on a connected finite-dimensional manifold {X}.

The reduction to the {p}-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 {X} to be close to the identity. I record this argument (which appears for instance in this paper of Lee) below the fold also.

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