This is probably a really basic question (hence my asking it here as opposed to MO). In a comment to a question on mathoverflow (https://mathoverflow.net/questions/121168/group-of-diffeomorphisms-of-a-manifold), Allen Knutson wrote, "You can use an element of $\mathit{Diff}(S^n)$ to glue together two $(n+1)$-balls. E.g. $\mathit{Diff}(S^6)$ has 28 components." Liviu Nicolaescu then added, "the connected components of $\mathit{Diff}(S^n)$, 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 $\mathit{Diff}$ give rise to symplectic structures, or how the geometry of $\mathit{Diff}$ accounts for whether the symplectic structure induced will be exotic.
Thanks!
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 $k > 1$.