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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.