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What are some examples of groups that can not be given a smooth structure such that they become a Lie Group?

Edit: To be a bit more specific, I was hoping that somebody could give an example of a finite dimensional topological group that is a topological manifold but does not admit a smooth structure making it into a Lie Group.

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    $\begingroup$ $\mathbb{R}^\infty$ $\endgroup$
    – Jimmy R
    Jun 13, 2015 at 16:58

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Any group is a Lie group if you give it the discrete topology. The better question is whether a topological group has a smooth structure that makes it a Lie group.

Local compactness is obviously necessary (because you want finite dimensions), so any non-locally compact group will be an example.

Generally, most locally compact groups are Lie groups. This question is essentially Hilbert's 5th Problem, which has been solved: https://en.wikipedia.org/wiki/Hilbert's_fifth_problem

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    $\begingroup$ Why should most locally compact groups be locally Euclidean? I would have trouble thinking of more than, say, the p-adics, but this seems more like a personal failure than a statement that most groups really are manifolds. $\endgroup$
    – user98602
    Jun 13, 2015 at 17:20
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    $\begingroup$ As @MikeMiller said, you really need to add the requirement that the group has the homeomorphism type of a manifold to exclude profinite groups and so forth. $\endgroup$
    – Jim Belk
    Jun 13, 2015 at 19:22
  • $\begingroup$ Are there topological groups that are topological manifolds that cannot be made into smooth manifolds? $\endgroup$
    – Memeozuki
    Jun 14, 2015 at 0:52
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    $\begingroup$ @Abraham: no. This is part of the solution to Hilbert's fifth problem. See, for example, terrytao.wordpress.com/2011/10/08/…. $\endgroup$
    – Qiaochu Yuan
    Jun 14, 2015 at 3:25
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You can take any uncountable Cartesian product of rationals $Q$ with discrete topology. This have discrete topology again (pre-images of $\emptyset \times \emptyset \times \cdots \times \{x\} \times \emptyset \times \emptyset \cdots, x\in Q$) are one-point elements in the product). Thus it has dimension 0 as a topological manifold. It is a group. It is a topological group (addition is bi-continuous and inversion is continuous, because of the discrete topology). Moreover, it is locally compact. But it is not second countable.

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