In exercise 4-5 from John M. Lee's "introduction to smooth manifolds" I have proven that there is a surjective smooth submersion from $\mathbb C^2\setminus\{0\}$ to $\mathbb{CP}^1$, namely the map $\pi(z_1,z_2) = [z_1:z_2]$. The next exercise asks to show that $\mathbb{CP}^1$ is diffeomorphic to the sphere $\mathbb S^2$.
Theorem 4.31 states that if $M, N_1$ and $N_2$ are smooth manifolds, and $\pi_1 \colon M\to N_1$ and $\pi_2\colon M\to N_2$ are surjective smooth submersions that are constant on each other's fibers, then there exists a unique diffeomorphism $F\colon N_1\to N_2$ such that $F\circ \pi_1 = \pi_2$.
If possible I would like to use this theorem, because I already have a surjective smooth submersion from $\mathbb C^2 \setminus \{0\}$ to $\mathbb{CP}^1$. However, I need help with finding a good candidate map from $\mathbb{C}^2\setminus \{0\}$ to $\mathbb S^2$. Do you have any suggestions? Or do you have an argument why this would not work. Thanks in advance!
As Jason DeVito states one can use the map $\mathbb C^2 \setminus 0\to \mathbb S^3:x\mapsto \frac{x}{|x|}$ followed by the map $$ \mathbb S^3 \to \mathbb S^2 : (w,z) \mapsto \left( z\overline{w}+ w\overline{z}, iw\overline{z}-iz\overline{w}, z\overline{z}-w\overline{w} \right) $$ as defined in exercise 2-3(c) of Lee's introduction to smooth manifolds.