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I'm trying to prove that the projective plane $\mathbb{P}^n$ is orientable is and only if $n$ is odd. To do that that, I have a hint,to prove that the antipodal map is orientation preserving if only if $n$ is odd, I've done that, but it don't know how to conclude the result.

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    How to you define $\mathbb P^n$?2012-05-20
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    Identifying antipodal points on the sphere...2012-05-20
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    What do you know about the orientability of the sphere?2012-05-20
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    I know that it is a orientable manifold...2012-05-20
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    So, what happens if you simply take the orientation of the sphere as orientation of the projective space?2012-05-20
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    @Phira what do you mean?2012-05-21
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    I think the word "plane" is misleading if we aren't talking about the 2-dimensional object; perhaps projective "space" would be a better term?2012-05-21

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