# orientability of the möbius strip using homology

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.

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

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 $H_2(M,\partial M)=0$.