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.
Projective Space orientation
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differential-geometry
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0How to you define $\mathbb P^n$? – 2012-05-20
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0Identifying antipodal points on the sphere... – 2012-05-20
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0What do you know about the orientability of the sphere? – 2012-05-20
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0I know that it is a orientable manifold... – 2012-05-20
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0So, what happens if you simply take the orientation of the sphere as orientation of the projective space? – 2012-05-20
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0@Phira what do you mean? – 2012-05-21
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1I 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