Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, those there exist a topological property $X$, such that a compact group $G$ is Lie IFF it satisfies $X$? I recall hearing that $X$ can be given in terms of connectedness . . . but can't find a mention of this in the literature.
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10$\begingroup$ Yes: iff it is a topological manifold (or equivalently if there exists a neighborhood of $1$ that is homeomorphic to $\mathbf{R}^n$ for some $n$). I think this can be derived from the Peter-Weyl theorem. It's also true (and harder) in the locally compact case (Gleason, Yamabe, Montgomery-Zippin). $\endgroup$– YCorCommented Jul 5, 2019 at 19:24
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7$\begingroup$ BTW "it is Lie" means precisely "there is a differentiable structure making it a Lie group" but it turns out that this differentiable structure is unique. This follows from the fact that every isomorphism of topological groups between Lie groups is actually differentiable. Also in the latter sentence, "differentiable" can mean $C^\infty$-differentiable (by default), but also $C^k$-differentiable for any $k\ge 1$, and also real analytic. $\endgroup$– YCorCommented Jul 5, 2019 at 19:49
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$\begingroup$ Isn't there also a description (via one of Hilbert's problems) in terms of having no small subgroups? EDIT: Ah, I guess I misremembered the generality; according to Wikipedia, this applies to LC groups already known to be projective limits of Lie groups. $\endgroup$– LSpiceCommented Jul 5, 2019 at 23:56
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$\begingroup$ A compact group is a Lie group if and only if it is locally contractible [Theorem 10.80 in the 4th edition of Hofmann-Morris, Structure of compact groups]. The theorem lists several more equivalent topological conditions. $\endgroup$– LinusCommented Aug 9, 2022 at 7:52
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