I saw from some algebraic topology paper like J.F. Adams' vector field on sphere claiming that $\mathbb{RP}^n/\mathbb{RP^{n-1}}$ is homeomorphic to $S^n$. So, I have a similar question regarding $\mathbb{CP}^n/\mathbb{CP}^{n-1}$. Is it just $S^{2n}$?
I hope someone can have a more pictorial explanation on that!
Thanks in advance for answering.
$\mathbb{CP}^n$ is homeomorphic to a CW complex with a single cell in each even dimension between $0$ and $2n$ (inclusive), i.e. $\mathbb{CP}^n = e^0 \cup e^2\cup\dots\cup e^{2n-2}\cup e^{2n}$. The union of the cells up to and including dimension $2k$ can be identified with $\mathbb{CP}^k \subset \mathbb{CP}^n$. When we quotient out by such a subspace, all these cells collapse to a single point, so $\mathbb{CP}^n/\mathbb{CP}^k = e^0\cup e^{2k+2}\cup e^{2k+4}\cup\dots e^{2n-2}\cup e^{2n}$. In particular, $\mathbb{CP}^n/\mathbb{CP}^{n-1} = e^0\cup e^{2n}$ which is a CW complex homeomorphic to $S^{2n}$.
Said another way, we can obtain $\mathbb{CP}^n$ from $\mathbb{CP}^{n-1}$ by attaching a $2n$-cell. When we quotient by $\mathbb{CP}^{n-1}$, the resulting space is obtained by attaching a $2n$-cell to a point, which results in a sphere of dimension $2n$.