The question is pretty straight-forward, but I have been unable to find an answer:
Let $G$ be a (compact Hausdorff) Lie group. If $X$ is homeomorphic to $G$ as a topological space, is $X$ then also a topological manifold?
2026-04-04 00:37:04.1775263024
If $X$ is homeomorphic to a Lie group, is $X$ is a manifold?
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Not sure if this fully answers the question, but might point in some useful direction. Per wikipedia:
"Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same." (https://en.wikipedia.org/wiki/Homeomorphism)
and then from the definition of a Lie group: "Lie groups, named after Sophus Lie, are differentiable manifolds that ..." (https://en.wikipedia.org/wiki/Manifold).
Combining these two pieces of information seems to suggest that $X$ is indeed a manifold.
I hope this helps.