We know that there exist exotic smooth structures on Euclidean space on $\mathbb{R}^4$, say. On the other hand, if we require the Lie group $G$ to be isomorphic to $\mathbb{R}^n$ as a topological group, I suspect (though I can't prove) that $G$ must have the same smooth structure as $\mathbb{R}^n$. But when we simply require that there be a homeomorphism, I can't think of any tool that would be useful to proving that $G$ does not have an exotic smooth structure.



Is there any exotic Lie Group homeomorphic to $\mathbb{R}^n$?
By [this link](https://books.google.de/books?id=gRTDO-wVhj0C&pg=PA49&redir_esc=y#v=onepage&q&f=false), every continuous homomorphism between Lie groups is smooth, so the isomorphism of topological groups is actually a diffeomorphism. In particular, $G$ has the same smooth structure as $\mathbb R^n$.


