Timeline for Describing compact Lie groups in purely topological terms
Current License: CC BY-SA 4.0
6 events
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| Aug 9, 2022 at 7:52 | comment | added | Linus | 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. | |
| Jul 5, 2019 at 23:56 | comment | added | LSpice | 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. | |
| Jul 5, 2019 at 20:42 | history | edited | Max Schattman | CC BY-SA 4.0 |
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| Jul 5, 2019 at 19:49 | comment | added | YCor | 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. | |
| Jul 5, 2019 at 19:24 | comment | added | YCor | 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). | |
| Jul 5, 2019 at 19:21 | history | asked | Max Schattman | CC BY-SA 4.0 |