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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?

D_S
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Horned Sphere
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  • Why wouldn't it? Isn't every Lie group a smooth manifold? – D_S Jul 15 '22 at 15:40
  • 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? – Horned Sphere Jul 15 '22 at 15:46
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    I now understand your question and have edited it to clarify what I think you want to know. – D_S Jul 15 '22 at 16:20
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    I guess there could be a question about whether the continuity of the group operations inescapably implies their smoothness... – paul garrett Jul 15 '22 at 16:36
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    Your question is unclear. When you say "If is a topological manifold, and I can show that 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. – Moishe Kohan Jul 15 '22 at 16:42
  • ... while in the 1st case, there are counter-examples, although, not among simple real Lie groups. – Moishe Kohan Jul 15 '22 at 22:23

1 Answers1

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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.

Leonid
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  • 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. – Filippo Jul 15 '22 at 15:59
  • 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. – Leonid Jul 15 '22 at 16:06
  • $$\text{Lie group = set + smooth manifold structure + group structure}$$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...) – Filippo Jul 15 '22 at 16:33
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    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. – Leonid Jul 15 '22 at 16:49
  • See also this answer: https://math.stackexchange.com/a/1606915/532409 – Quillo Sep 21 '23 at 09:40