How many smooth structures are there on $S^2$, $S^3$, and $S^4$ up to diffeomorphism? I looked around and couldn't find an answer; two books I have say different things on the subject.
I know one of these should still be an open question.
2026-04-03 02:01:30.1775181690
Non-diffeomorphic structures on the sphere
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All manifolds of dimension up to 3 have a unique smooth structure up to diffeomorphism. 1 is essentially trivial, 2 is due to Rado, 3 is due to Moise.
Whether or not there exist exotic smooth structures on $S^4$ is wide open. This is known as the smooth Poincaré conjecture in 4 dimensions. Some topologists think there should even be infinitely many exotic structures on $S^4$ but this opinion is certainly not uniform.
The keyword you want is exotic sphere.