For $n > 1$, $2n$-dimensional sphere $S^{2n}$ does not admit symplectic structures. Then how about the product with a manifold? Are there any results about the symplectic structures on $M \times S^{2n}$?



Symplectic structures on $M \times S^{2n}$
From the [Künneth theorem](http://en.wikipedia.org/wiki/K%25C3%25BCnneth_theorem#Singular_homology_with_coefficients_in_a_field) you can check that there is no class $\omega\in H^2(M^{2d}\times S^{2n};\mathbb{R})$ such that $\omega^{d+n}\neq 0$. (This is an excellent thing for you to work out for yourself.) Since a symplectic form is closed and nondegenerate, this shows that no symplectic structure on $M\times S^{2n}$ can exist for $n>1$ if $M$ is compact.


(Thanks to Eric and David for pointing out that I was assuming $M$ to be compact; this is necessary so that nondegeneracy implies that the top power of $\omega$ represents a nonzero multiple of the fundamental class. In his answer David Speyer gives a nice counterexample if $M$ is not required to be compact.)


