In Gompf's 1994 article, "Some new symplectic 4-manifolds," the author defines a connect sum operation for two symplectic manifolds of the same dimension along a common codimension-2 symplectic submanifold. The construction requires a symplectic involution of the open 2-dimensional annulus of radius $\epsilon$, which can be given explicitly in polar coordinates:
$f(r,\theta)=(\sqrt{\epsilon^2-r^2},-\theta)$
The author does not check that this is a symplectomorphism, but I think it is clear that it is, since it preserves the polar area form (right?).
One can ask why the codimension-2 specification is required for the construction. Evidently, this is because a codimension-2k symplectic sum construction would require a symplectic involution of the k-dimensional annulus. If such a symplectomorphism existed, I can see how it could be used to patch two 2k-dimensional balls to put a symplectic structure on the 2k-sphere, which is impossible when k>1 for cohomological reasons.
My question: Is there a simple geometric reason why the natural involutive diffeomorphism of the 2k-dimensional annulus $A$ generalizing $f$ (informally, this diffeomorphism would turn $A$ inside-out and compose with the antipodal map) is not a symplectomorphism of $A$, for appropriate choice of symplectic structure on $A$?
I think the a $\textit{geometric}$ reason is that the existence of such a diffeomorphism implies that the two annuli can be glued together symplectically thus giving a symplectic form on $S^n$. By standard de Rham theory, only $S^2$ can carry such a symplectic form.