I recently got a statement like this, $\frac{\Bbb RP^n}{\Bbb RP^{n-2}}$ is homotopically equivalent to $\Bbb S^n\lor\Bbb S^{n-1}$ if and only if $n$ is odd.
If $n$ is even then using long exact sequence for homology to the pair $(\Bbb RP^n,\Bbb RP^{n-2})$ and $H_k(\Bbb RP^n,\Bbb RP^{n-2})\cong H_k\big(\frac{\Bbb RP^n}{\Bbb RP^{n-2}}\big)$ and $H_k(\Bbb S^n\lor\Bbb S^{n-1})\cong H_k(\Bbb S^n)\oplus H_k(\Bbb S^{n-1})$ we have one direction. But I have a problem with other direction, namely if $n$ is odd then $\frac{\Bbb RP^n}{\Bbb RP^{n-2}}$ is homotopically equivalent to $\Bbb S^{n}\lor\Bbb S^{n-1}$. Any help will be appreciated.