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If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being closed, that this is the crux.

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    The *symplectic-geometry* tag is a bit cryptic here :) But this is connected to the proof that $S^4$ is not a symplectic manifold.2012-05-06

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