TITLE: Diffeomorphism of $\mathbb{C}P^1$ and $S^2$.
QUESTION [20 upvotes]: In an exercise I am asked to find a (smooth) submersion of $S^3$ onto the sphere $S^2$. So far I have a submersion of $S^3$ onto $\mathbb{C}P^1$.
Are $\mathbb{C}P^1$ and $S^2$ diffeomorphic? If so, how does one construct a diffeomorphism between the two and what is the geometric intuition behind the construction?
Thank you.

REPLY [4 votes]: The explicit map you are looking for is the following. Write 
$$S^2=\big\{(z,\alpha) \in \mathbb{C} \times \mathbb{R} \, \big| \, |z|^2+\alpha^2=1\big\}.$$
Then, employing the standard homogeneous coordinates on projective space, we have that the map
\begin{align*}
\varphi: S^2 & \to \mathbb{CP}^1 \\
(z,\alpha) & \mapsto [2z:1-\alpha]
\end{align*}
is a diffeomorphism.