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For which $n$ there exists a closed oriented manifold $M$ of dimension $2n$ which satisfies the following Betti number condition?

$$ \beta_0=\beta_n=\beta_{2n}=1;\ \beta_i=0,\ i\ne0,n,2n. $$

By looking at the projective planes $\mathbb{CP}^2,\mathbb{HP}^2,\mathbb{OP}^2$ (The last one is the Cayley plane, for which existance is nontrivial. It has a cell complex structure with only one cell in dimension $0,8,16$ via the Hopf map), we get examples when $n=2,4,8$. And by this MSE question, $n$ is never an odd number since in that case the pairing in the middle cohomology should be skew-symmetric.

So for even $n$, are $2,4,8$ all possible answers? Is this question somehow turns into the classification of the division algebras? To be a little more curious, does our construction above cover all such possible manifolds?

cybcat
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    These manifolds are known as rational projective planes, see this related question. Note, $n$ must be even (as is assumed in that question, $n = 2d$) because for $n$ odd, the cup product is symplectic on $H^n$ and hence $b_n$ is even. – Michael Albanese Aug 14 '24 at 06:05
  • I wonder how much the answer depends on the coefficient field used to specify the Betti numbers. The link in Michael Albanese's comment addresses the case of $\mathbb{Q}$, but what about "mod $p$" projective planes? – JHF Aug 14 '24 at 15:55

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