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.
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1$\begingroup$ A connected $n$-manifold is orientable iff $H_n(M,\partial M)\cong\mathbb Z$. $\endgroup$– Cheerful ParsnipMay 16, 2012 at 12:26
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$\begingroup$ how can one prove this? or is it in hatcher's book, at whic page? $\endgroup$– pascalMay 16, 2012 at 12:32
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4$\begingroup$ See Theorem 3.43 of Hatcher. $\endgroup$– Cheerful ParsnipMay 16, 2012 at 12:42
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$\begingroup$ @Jim - Surely this should be an answer? :) $\endgroup$– Juan SMay 17, 2012 at 0:15
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1 Answer
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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 $H_2(M,\partial M)=0$.
answered May 17, 2012 at 3:06