# Groups that are not Lie Groups

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.

• $\mathbb{R}^\infty$ Jun 13, 2015 at 16:58

## 2 Answers

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

• 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.
– user98602
Jun 13, 2015 at 17:20
• 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. Jun 13, 2015 at 19:22
• Are there topological groups that are topological manifolds that cannot be made into smooth manifolds? Jun 14, 2015 at 0:52
• @Abraham: no. This is part of the solution to Hilbert's fifth problem. See, for example, terrytao.wordpress.com/2011/10/08/…. Jun 14, 2015 at 3:25

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.