Recall that a *Lie group* is a group object in the category of *C*∞ manifolds.


If I have a group object in the category of topological manifolds, can I necessarily equip it with a smooth structure so that all the group operations are smooth? If so, how unique is this structure?


Is a continuous group-homomorphism between two Lie groups necessarily smooth?



Is every  group object in TopMan a Lie group?
As Greg Kuperberg indicated in the comments, this is Hilbert's 5th Problem. The answer is yes, a theorem of Gleason, Montgomery and Zippin from the 1950's.


