4
$\begingroup$

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?

$\endgroup$
6
$\begingroup$

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 $M$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.