Which are homotopy classes of mappings $\mathbb{CP}^n \to \mathbb{CP}^m$ for $n < m$?
In real case, even for any cellular complex $X$ with $\dim X<m$ homotopy classes of mappings $X \to \mathbb{RP}^m$ are in bijection with $H^1 (X, \pi_1(\mathbb{RP}^m)=\mathbb Z_2)$: the reason is that universal covering $S^m \to \mathbb{RP}^m$ has only cells of dimensions 0 and m.
Observe that any map $\mathbb{CP}^{n} \rightarrow \mathbb{CP}^{\infty}$ can be pushed down to the $2n$-skeleton of $\mathbb{CP}^{\infty}$, due to cellular approximation.
This establishes a bijection between homotopy classes of maps
$[\mathbb{CP}^{n}, \mathbb{CP}^{\infty}] \simeq [\mathbb{CP}^{n}, \mathbb{CP}^{m}]$
for all $m \geq n$. The left hand side now admits a multitude of descriptions, one of them being that it is $H^{2}(\mathbb{CP}^{n}, \mathbb{Z})$, as $\mathbb{CP}^{\infty}$ is the classifying space for complex line bundles.
