Suppose $m > n$, and let $f : \mathbb CP^m \to \mathbb CP^n $ be continuous, the claim is that the induced map between the homology(over $\mathbb Z$) is zero.
I have no clue why this should be true. The only thing I know about $\mathbb CP^n$ is from Hatcher that it has a CW-complex structure, so I might as well use it. To prove this I only need to show that for $0 \leq 2k \leq n$, $f_{\ast}:H_{2k}\mathbb CP^m \to H_{2k}\mathbb CP^n$, is zero. Note a generator for $H_{2k}CP^m$ is simply the $2k-cell$ i.e. $\delta:\Delta^{2k} \to e^{2k}$ where $e^{2k}$ is the open ball. Then it is mapped to $f \circ \delta$. Intuitively speaking that if I want to map $\mathbb CP^m \to \mathbb CP^n$, I wound need to "project things down" or squeeze them to things with lower dimensions, meaning that $\delta \circ f$ can no longer be a $2k$ dimensional open ball.
How do I make this precise mathematically though?