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I need the examples of closed oriented even-dimensional manifolds with all central even degree cohomology groups ( field is rational numbers) are zero. First and last non-zero Betti numbers are always one. Simplest examples are the product of two odd dimensional spheres or even dimensional homology spheres or closed oriented surfaces. In all these examples the cohomology is simple. I need some more complicated examples of such kinds. The cohomology ring of these type of manifolds easily understandable due to odd degrees central Betti numbers and perfect pairing ( due to fundamental class).

I consider the only 2d-dimensional closed connected oriented manifolds. $\beta_{0}$ and $\beta_{2d}$ of 2d-manifold are always one. I mean except these two Betti numbers all others even degree Betti numbers are zero. Odd degree Betti numbers are may be zero or not.

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  • $\begingroup$ I am not sure what you mean. Are you looking for closed, oriented manifolds with all even Betti numbers equal to zero? $\endgroup$ – Michael Albanese Mar 25 at 13:49
  • $\begingroup$ "Closed oriented surfaces"? It seems to me that the torus is such a thing, but its central homology is nontrivial. Presumably the Poincare homology sphere is another example; I'm not sure why you restricted to "even-dimensional homology spheres". Perhaps you can clarify your question a bit more. $\endgroup$ – John Hughes Mar 25 at 14:31
  • $\begingroup$ sorry, you are right I mean even degree central cohomology groups are zero. $\endgroup$ – King Khan Mar 25 at 15:15
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    $\begingroup$ I consider the only even dimensional manifolds. $\endgroup$ – King Khan Mar 25 at 15:23
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Take n-times connected sum of the product of two odd dimensional spheres with itself. This is similar to every closed oriented surface obtain by the connected sum of a torus with itself.

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