Homotopically equivalent closed non-orientable manifolds with different dimensions.

Question

I know that the two closed orientable even-dimensional manifolds of different dimensions are not homotopically equivalent due to the fundamental classes of cohomology rings. Let $M$ and $N$ be the closed non-orientable even-dimensional manifolds of different dimensions. is it possible that $M$ and $N$ are homotopically equivalent? Moreover, what are the rational cohomology groups of $M$ and $N$ in this case?

Answer 1

No, since if they were homotopy equivalent then their homology $H_*(M,\Bbb Z/2\Bbb Z)$ with coefficients in $\Bbb Z/2\Bbb Z$ would be isomorphic. But the top homology group $H_n(M,\Bbb Z/2\Bbb Z)$ occurs when $n$ is the dimension of the manifold, even for non-orientable $M$.