1
$\begingroup$

I read in Hatcher's "Algebraic topology" book about orientability of topological maifolds using homology. now I would like to know how one can apply this to show that the möbius strip is not orientable? i have no idea.

$\endgroup$
4
  • 1
    $\begingroup$ A connected $n$-manifold is orientable iff $H_n(M,\partial M)\cong\mathbb Z$. $\endgroup$ – Cheerful Parsnip May 16 '12 at 12:26
  • $\begingroup$ how can one prove this? or is it in hatcher's book, at whic page? $\endgroup$ – pascal May 16 '12 at 12:32
  • 4
    $\begingroup$ See Theorem 3.43 of Hatcher. $\endgroup$ – Cheerful Parsnip May 16 '12 at 12:42
  • $\begingroup$ @Jim - Surely this should be an answer? :) $\endgroup$ – Juan S May 17 '12 at 0:15
1
$\begingroup$

One can show that a connected $n$-manifold is orientable iff $H_n(M,\partial M)\cong \mathbb Z$. Theorem 3.43 of Hatcher gives the "only if" direction. Namely, if $M$ is orientanble, then $H_n(M,\partial M)\cong H^0(M)\cong \mathbb Z$. So to show that the Möbius strip is not orientable, it suffices to show $H_2(M,\partial M)=0$.

$\endgroup$

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.