Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?
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This is definitely not my expertise, but here is a stab at it.
A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group.
On the other hand, there are exotic structures on Lie groups (including compact ones).
So take a Lie group $G$ that admits an exotic structure. Let $M$ denote the exotic version of $G$. Then $G$ and $M$ are homeomorphic but not diffeomorphic. However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.
answered Jun 20, 2016 at 20:50
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6$\begingroup$ You don't need the full strength of Hilbert's 5th problem, just the simpler result that a continuous homomorphism between Lie groups is automatically smooth. $\endgroup$ Commented Jun 20, 2016 at 20:55
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5$\begingroup$ I think we do not need the full solution to Hilbert's 5t problem. It is elementary that a continuous homomorphism of Lie groups is automatically smooth. So a second countable locally Euclidean group can have at most one differentiable structure making it into a Lie group (cf. a book like Warner's). Any exotic structure on a Lie group will do, like you said. for instance in $R^4$. $\endgroup$ Commented Jun 20, 2016 at 20:57