Can topological groups be smoothed into lie groups?

Question

I've been thinking about this for the past couple of days and I'm really not sure of the answer..
By "smoothed" I mean that for any arbitrary precision we can find a Lie group which approximates the topological group within the precision, much in the same way we can find an arbitrary smooth approximation of a topological manifold.
I'm no expert in the theory of Lie groups however so I'm not sure if this idea can be extended.
Does anyone know the answer to this?

Answer 1

Your question seems to be a version of Hilbert's fifth problem. There are some obvious necessary conditions for a topological group to be a Lie group: it should be locally compact Hausdorff. Somewhat more subtly, there should be a neighborhood of the identity that does not contain any non-trivial subgroup. It is a famous theorem usually attributed to (at least!) Gleason, Montgomery-Zippin, and Yamabe that the converse of this statement is also true: a locally compact Hausdorff topological group such that there is a neighborhood of the identity not containing a non-trivial subgroup is a Lie group. I like Terry Tao's book

http://www.ams.org/bookstore-getitem/item=gsm-153

as a reference for and introduction to this topic.