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.
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.