Suppose $X$ be a CW complex. Then we know that, $$\frac{X^n}{X^{n-1}}=\underset{\text{$\sigma$ an $n$-cell}}{\large\lor} \Bbb S^n,$$ Where $X^n$ is $n$-th skeleton of $X$ and $X^{n-1}$ is $(n-1)$-th skeleton of $X$ and equality in sense of homeomophism.
My question is is there any well known space to write the quotient $X^n \big/ X^k$, when $1\leq k\leq (n-2)$.
Recently I found a particular case namely, $\Bbb S^{2n+1}\big/\Bbb S^1=\Bbb CP^{n}$ (Homeomophic).
Not really. The space $X^n / X^k$ can be any CW-complex with exactly one $0$-cell and all other cells in dimensions $k+1, \dots, n$.