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even. But, then you need de Rham's theorem, universal coefficients theorem, and the fact that Q is an injective object in the abelian category of Z-modules. I'm highly distrusting of the second proof give too. The fact is you need to consider the pushforward of the continuously oriented local frames on the sphere. The proof is not short nor is it trivial.
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You cannot use homology theory to prove that real projective space is orientable for n odd. The theorem is that if a compact manifold is orientable, then its top de rham cohomology group is nonzero. By a sketchy argument of sorts, you can use the contrapositive to show that it is not orientable for n
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there is a nice mention of the covering map S → RP, and the fact that these are Lie groups. So it seems a bit odd to omit mention of the even simpler double cover S → RP. (And why not also mention the n-fold covers S → RP and their (its) Lie group status as
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You wouldn't put the constraints on v inside the set braces, since v doesn't vary. So you'd write: "the equivalence class of v \epsilon R^{n+1}-{0} is = \{ \lambda v: \lambda \epsilon R^* \}".
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could the equivalence class be written like this: = \{ \lambda v: v \epsilon R^{n+1} -{0} \and \lambda \epsilon R^* \}
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in? Such as the Euler characteristic and homology groups?
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