# Non-diffeomorphic structures on the sphere

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.

• I would be interested to know what your two books say so that I can clarify them. – user98602 Dec 3 '15 at 17:14
• 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. – Dac0 Dec 3 '15 at 17:33

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.