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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.

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    $\begingroup$ I would be interested to know what your two books say so that I can clarify them. $\endgroup$ – user98602 Dec 3 '15 at 17:14
  • $\begingroup$ I think that simply one of them has an error of editing since it gives $S^2->1$, $S^3->?$ and $S^4->1$. Your answer was what I needed, thank you. $\endgroup$ – Dac0 Dec 3 '15 at 17:33
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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.

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