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What is an example of an orientable compact $2n$ dimensional manifold $M$ whose all even dimensional De Rham cohomology groups $H_{\mathrm{DeR}}^{2i}(M)$ are nonzero, but $M$ does not admit any symplectic structure?

Added: As it is indicated in the comments, this post is not a duplicated post.

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    $\begingroup$ $S^2\times S^4$? $3\mathbb{CP}^2$? $\endgroup$
    – Marco Golla
    Commented Oct 13, 2018 at 21:42
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    $\begingroup$ Just to mention that examples can't be surfaces, as mentioned in math.stackexchange.com/questions/2116869/… $\endgroup$
    – YCor
    Commented Oct 13, 2018 at 21:43
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    $\begingroup$ The linked question (claimed duplicate) does not refer to the nonvanishing of the even De Rham cohomology, and precisely the answer to this question is this nonvanishing. So the answer to the linked question does not answer the current question, so it's not a duplicate. $\endgroup$
    – YCor
    Commented Oct 13, 2018 at 22:18
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    $\begingroup$ p12 of math.tecnico.ulisboa.pt/~xvi-iwgp/talks/ACannas.pdf, it is mentioned that the 4-manifold $(S^2\times S^2)\# (S^2\times S^2)$ admits no almost complex (and hence no symplectic) structure. Its $b_2$ is clearly nonzero, and hence it satisfies the condition on even Betti numbers. $\endgroup$
    – YCor
    Commented Oct 13, 2018 at 22:25
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    $\begingroup$ @ChrisGerig This gives one example in each dimension multiple of 4, right? You probably know examples in dimension $4n+2\ge 6$ as well? $\endgroup$
    – YCor
    Commented Oct 15, 2018 at 21:52

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For your $n=2k$, $\mathbb{C} P^n\#\mathbb{C} P^n$ does not even admit an almost complex structure, so it cannot be symplectic.

See also: 1) Goertsches-Konstantis' paper "Almost complex structures on connected sums of complex projective spaces" which answers the following MO question
2) Does $\Bbb{CP}^{2n} \# \Bbb{CP}^{2n}$ ever support an almost complex structure?

Including Miller's comment for the remaining (your $n=2k+1$) dimensions: $S^2\times(S^4\times\cdots\times S^4)$

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