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It is said that $\mathbb R P^n \to \mathbb R P^n/\mathbb R P^{n-1}$ induces isomorphism on $H_n$ when $n$ is odd. How can we prove this?

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Compute the cellular homology groups of both, check they're isomorphic. Look at the proof that cellular homology computes singular homology and see that it implies exactly what you want. –  Ryan Budney Apr 4 '12 at 15:38
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