Note: this isn't homework or anything. Just trying to sort out some basics.
I'm trying to understand some examples of induced maps on $\mathbb{C}P^n$. In particular, say we have a map $f:\mathbb{C}P^n \rightarrow \mathbb{C}P^n$ that swaps homogenous coordinates, like $f(a_0: a_1: a_2: a_3) = f(a_1: a_0: a_2: a_3)$. First of all, I'm having a hard time visualizing what this "swapping" actually does to an element in $\mathbb{C}P^n$. And how would I figure out the induced map on homology? My thoughts were maybe to use that $S^{2n+1}$ is a cover for $\mathbb{C}P^n$ and make some kind of commutative diagram, since induced maps on spheres are easier to understand.