3 corrected an error ; added 1 characters in body

I would develop a little the remark of Spiro Karigiannis on the topological obstruction for a manifold to have a symplectic structure.

Two first rough necessary conditions for the existence are: the manifold has to be even-dimensional and orientable.

Apart from this, we have also the cohomology ring condition:

If a connected compact $2n$-dimensional manifold $M$ has a symplectic structure, then there exists $u\in H_{dR}^2(M)$ such that $u^n\neq u^k\neq 0\in H_{dR}^{2n}(M)$, H_{dR}^{2k}(M)$for$k=1,\ldots,n$, and in particular$H_{dR}^{2n}(M)\neq 0$.H_{dR}^{2k}(M)\neq 0$,for $k=1,\ldots,n$.

Proof. For It is sufficient to establish that, for any symplectic form $\omega$ on $M$, we have $[\omega]^n=[\omega^n]\neq 0$. Infact, because Because of the nondegeneracy of $\omega$, we have that $\omega^n$ is a volume form, so its integral over $M$ is not zero and hence $[\omega^n]\neq 0$.

This condition and the computation $H_{dR}^{2n}(S^{2n})=0$ H_{dR}^{2}(S^{2n})=0$for any$n>1$imply the nonexistence of symplectic structures on$S^{2n}$for all$n>1$. 2 added a single dollar sign. I would develop a little the remark of Spiro Karigiannis on the topological obstruction for a manifold to have a symplectic structure. Two first rough necessary conditions for the existence are: the manifold has to be even-dimensional and orientable. Apart from this, we have also the cohomology ring condition: If a connected compact$2n$-dimensional manifold$M$has a symplectic structure, then there exists$u\in H_{dR}^2(M)$such that$u^n\neq 0\in H_{dR}^{2n}(M)$, and in particular$H_{dR}^{2n}(M)\neq 0$. Proof. For any symplectic form$\omega$on$M$, we have$[\omega]^n=[\omega^n]\neq 0$. Infact, because of the nondegeneracy of$\omega$,$\omega^n$is a volume form, so its integral over$M$is not zero and hence$[\omega^n]\neq 0$. This condition and the computation$H_{dR}^{2n}(S^{2n})=0$for any$n>1$imply the nonexistence of symplectic structures on$S^{2n}$for all$n>1$. 1 I would develop a little the remark of Spiro Karigiannis on the topological obstruction for a manifold to have a symplectic structure. Two first rough necessary conditions for the existence are: the manifold has to be even-dimensional and orientable. Apart from this, we have also the cohomology ring condition: If a connected compact$2n$-dimensional manifold$M$has a symplectic structure, then there exists$u\in H_{dR}^2(M)$such that$u^n\neq 0\in H_{dR}^{2n}(M)$, and in particular H_{dR}^{2n}(M)\neq 0$.

Proof. For any symplectic form $\omega$ on $M$, we have $[\omega]^n=[\omega^n]\neq 0$. Infact, because of the nondegeneracy of $\omega$, $\omega^n$ is a volume form, so its integral over $M$ is not zero and hence $[\omega^n]\neq 0$.

This condition and the computation $H_{dR}^{2n}(S^{2n})=0$ for any $n>1$ imply the nonexistence of symplectic structures on $S^{2n}$ for all $n>1$.