Closed oriented even dimensional manifold with only three non-zero Betti numbers.

Question

The complex and quaternionic projective planes are the examples of a closed oriented even dimensional manifold with exactly three non-zero Betti numbers. For more example see the paper ''Rational analogs of projective planes''. In all these cases the middle Betti number is of even degree. Is there any closed oriented even dimensional manifold has exactly three non-zero Betti numbers with middle Betti number has odd degree?

Answer 1

Certainly. For instance, for any $n$ (even or odd), $S^n\times S^n$ is a closed orientable manifold of dimension $2n$ whose only nonzero Betti numbers are in degree $0,n$ and $2n$.