# Homotopically equivalent closed non-orientable manifolds with different dimensions.

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?

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 $$M$$.