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I have a little doubt about Lie Groups. If $M$ is a topological manifold, and I can show that $M$ has a Lie group structure (and therefore a smooth manifold structure), and the group operations are continuous with respect to the underlying topology of the topological manifold structure, does this imply that the topological manifold structure on $M$ is equivalent to the smooth manifold structure?

Is there some counter example for this?

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  • $\begingroup$ Why wouldn't it? Isn't every Lie group a smooth manifold? $\endgroup$
    – D_S
    Jul 15, 2022 at 15:40
  • $\begingroup$ So if I have a manifold and I must to prove tha this manifold is a smooth manifold. I just need to show it's a Lie group, rigth? $\endgroup$ Jul 15, 2022 at 15:46
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    $\begingroup$ I now understand your question and have edited it to clarify what I think you want to know. $\endgroup$
    – D_S
    Jul 15, 2022 at 16:20
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    $\begingroup$ I guess there could be a question about whether the continuity of the group operations inescapably implies their smoothness... $\endgroup$ Jul 15, 2022 at 16:36
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    $\begingroup$ Your question is unclear. When you say "If 𝑀 is a topological manifold, and I can show that 𝑀 has a Lie group structure" - do you mean that $M$ as a set has a Lie group structure or that $M$ as a topological space has a Lie group structure? In the second case, there is nothing to be proven. $\endgroup$ Jul 15, 2022 at 16:42

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A Lie Group is defined to be a smooth manifold which is also a group with compatible group operations. So a Lie Group is automatically a smooth manifold.

Edit: To elaborate a bit more: A Lie group has more structure than a smooth manifold, it has the extra group structure. So given a topological manifold, proving that it's a Lie group is an overkill if you just want to prove it's a smooth manifold. Lie group = smooth manifold + group.

It may be that OP hasn't learned the most general definition of Lie groups, like the one contained in Brian Hall's Lie groups & representations book, where he begins by defining a Lie group as a subgroup of the general linear group, but please be aware this the general linear group is already a smooth manifold hence this definition of Lie group already implicitly assumes a smooth manifold structure.

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  • $\begingroup$ The question of the OP suggests that the definitions are unclear, so I would be more precise: A (smooth) Lie group is not a smooth manifold, but given a Lie group we obtain a smooth manifold by removing the group operation from the given data. $\endgroup$
    – Filippo
    Jul 15, 2022 at 15:59
  • $\begingroup$ I'm not sure I agree with the statement "a Lie group is not a smooth manifold", since being a smooth manifold is part of the definition of being a Lie group. $\endgroup$
    – Leonid
    Jul 15, 2022 at 16:06
  • $\begingroup$ $$\text{Lie group = set + smooth manifold structure + group structure}$$Saying that a Lie group is a smooth manifold is like saying that a Lie group is a group (we all say that and that's okay as long as we remember that what we actually mean is that a Lie group without its smooth manifold structure is a group...) $\endgroup$
    – Filippo
    Jul 15, 2022 at 16:33
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    $\begingroup$ I feel like this is semantics at this point. To me, if a thing is x and y then in particular it is x and in particular it is also y. $\endgroup$
    – Leonid
    Jul 15, 2022 at 16:49
  • $\begingroup$ See also this answer: math.stackexchange.com/a/1606915/532409 $\endgroup$
    – Quillo
    Sep 21, 2023 at 9:40

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