Let $G$ be a compact connected metrizable group of finite topological dimension. Is $G$ a Lie group? Or what should be equivalent in this case, does it have a faithful finite-dimensional representation?

This is something that should be well known, but it is difficult to find.

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    $\begingroup$ The book From groups to geometry and back has some results, e.g., the theorem by Montgomery-Zippin, which might help. $\endgroup$ – Dietrich Burde Feb 29 at 10:10
  • $\begingroup$ @DietrichBurde Thanks! Montgomery-Zippin should imply that when $G$ is also locally connected, then - since it acts transitively in itself - it is a Lie group. Since conversely a compact connected Lie group is locally connected, my question now boils down to the problem whether there is a compact connected finite-dimensional metrizable group which is not locally connected. $\endgroup$ – user446046 Feb 29 at 11:28
  • $\begingroup$ @user446046 About "what should be equivalent": it's indeed equivalent: E. Cartan's theorem that every closed subgroup of $\mathrm{U}(n)$ is a Lie group, ensures that if a compact group has a faithful continuous finite-dim rep, then it's a Lie group, and the Peter-Weyl theorem ensures that compact groups have faithful finite-dimensional reps with arbitrary small kernel, and in particular compact Lie groups have faithful ones. $\endgroup$ – YCor Apr 11 at 22:19
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    $\begingroup$ About your comment: it's not true that a compact locally connected group is Lie: a counterexample is $(\mathrm{R}/\mathbf{Z})^I$ for $I$ infinite: it is not even locally contractible. Still it's true for finite-dimensional compact groups. $\endgroup$ – YCor Apr 11 at 22:23
  • $\begingroup$ @YCor Thanks for the example, I implicitly included to the assumptions in the comment above the finite-dimensionality, as in the original question. $\endgroup$ – user446046 Apr 15 at 13:56

This is false. As an example, consider a solenoid $G$, which is the inverse limit of, say, $2$-fold covering maps of circles $$ ... \to S^1\to S^1\to S^1. $$ This topological group has topological dimension 1, is metrizable, compact, connected, even abelian, but is not a Lie group (it is not locally connected). The local model for this group is the product of a Cantor set with an interval.

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