= Symplectic cohomology obstruction
{title2=$[\omega]^n\ne0$}
On a nonempty compact $2n$-dimensional <symplectic manifold> without boundary, with $n\geq1$, the class $[\omega]^n$ in top-degree <de Rham cohomology> is nonzero, since its integral in the <symplectic orientation> is positive. In particular $\omega$ cannot be an <exact differential form>: if $\omega=d\alpha$, then $\omega^n=d(\alpha\wedge\omega^{n-1})$ and the <Generalized Stokes theorem> would make that integral zero. Thus vanishing second <de Rham cohomology> obstructs a closed positive-dimensional <symplectic manifold>.
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