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This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow (, Allen Knutson wrote, "You can use an element of $Diff(S^n)$ to glue together two $(n+1)$-balls. E.g. $Diff(S^6)$ has 28 components." Liviu Nicolaescu then added, "the connected components of $Diff(S^n)$, if more than one, are responsible for the existence of exotic symplectic structures on $S^{n+1}$."

Basically, my question is: can someone explain this? I don't really see how elements of $Diff$ give rise to symplectic structures, or how the geometry of $Diff$ accounts for whether the symplectic structure induced will be exotic.


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up vote 4 down vote accepted

The comment has a typo. He means "exotic smooth structures." The Wikipedia page for exotic spheres has some basic information on this, especially this section.

The only sphere admitting a symplectic structure is $S^2$, since a symplectic manifold $(M, \omega)$ is even-dimensional and any symplectic form defines a nonzero deRham cohomology class $[\omega] \in H^2(M)$, but $H^2(S^{2k}) = 0$ for $k > 1$.

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Okay, that makes sense. Thanks! – Noah Schweber Feb 8 '13 at 20:24

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