diffeomorphism between the complex line and the sphere

Question

I want to write the diffeomorphism between the complex line and the sphere.

$$\mathbb{C}P^1 = \{<(z_0,z_1)>\ \vert\ (z_0,z_1) \ne 0\} \\S^2 = \{(x,y,z)\ \vert\ x^2+y^2+z^2 = 1\}$$

I get that you have to work using the stereographic projections

$$\alpha_N:S^2\backslash\{(0,0,1)\}\rightarrow \mathbb{C}:(x,y,z)\mapsto \frac{x+iy}{1-z} \\\alpha_S:S^2\backslash\{(0,0,-1)\}\rightarrow \mathbb{C}:(x,y,z)\mapsto \frac{x-iy}{1+z}$$

But how can I combine these two to find a diffeo $\phi$ from $S^2$ to $\mathbb{C}P^1$?

Intuitively I would write $\phi(x,y,z)=(\alpha_N(x,y,z),\alpha_S(x,y,z))$ but this makes no sense in $(0,0,1)$ and $(0,0,-1)$.

So maybe define

1. $\phi(x,y,z)=(\alpha_N(x,y,z),\alpha_S(x,y,z))$ if $(0,0,1)\ne(x,y,z)\ne(0,0,-1)$
2. $\phi(x,y,z)=(0,1)$ if $(x,y,z) = (0,0,-1)$
3. $\phi(x,y,z)=(1,0)$ if $(x,y,z) = (0,0,1)$

but this is not continuous.

Any other idea I have had is neither continuous neither bijective.

Any hint of how I can combine these two projections to make a diffeomorphism?

Answer 1

One thing to notice is that $\alpha_N(x,y,z) = \alpha_S(x,y,z)^{-1}$ (whenever both are defined). Indeed, this is a simple computation. \begin{equation} \alpha_N(x,y,z)\alpha_S(x,y,z) = \frac{(x + iy)(x - iy)}{(1 - z)(1 + z)} = \frac{x^2 + y^2}{1 - z^2} = 1 \end{equation} since $x^2 + y^2 + z^2 = 1$.

Now we define maps $f_N: S^2 - \{0,0,1\} \rightarrow \mathbb{C}P^1$ and $f_S: S^2 - \{0,0,-1\} \rightarrow \mathbb{C}P^1$ as \begin{equation} f_N(x,y,z) = (\alpha_N(x,y,z), 1) \end{equation} \begin{equation} f_S(x,y,z) = (1,\alpha_S(x,y,z)) \end{equation} Then the fact that $\alpha_N(x,y,z)\alpha_S(x,y,z) = 1$ implies that $f_N$ and $f_S$ coincide on the intersection of their domains (i.e. the sphere without the poles). This is because \begin{equation} (\alpha_N(x,y,z), 1) = \alpha_N(x,y,z)(1, \alpha_N(x,y,z)^{-1}) = \alpha_N(x,y,z)(1, \alpha_S(x,y,z)) \end{equation} and thus $f_N$ and $f_S$ give equivalent points on $\mathbb{C}P^1$ whenever both are defined.

Then $f_N$ and $f_S$ together give the desired diffeomorphism.