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I read somewhere that the hopf map can be expressed as $h(z_{1},z_{2})=\frac{z_{1}}{z_{2}}$ where $h:\mathbb{C}^{2}\rightarrow\mathbb{C}\cup\{\infty\}$.

I let $z_{1}=a+bi$ and $z_{2}=c+di$ and $h(z_{1},z_{2})=\frac{(ac+bd)+(bc-ad)i}{c^{2}+d^{2}}$

But the explicit formula for hopf map is $h(a,b,c,d)=(2(ad+bc),2(bd-ac),a^{2}+b^{2}-c^{2}-d^{2})$ and after stereographic projection $s(x,y,z)=\frac{x+iy}{1-z}$ I got $\frac{(ad+bc)+(bd-ac)i}{c^{2}+d^{2}}$ instead which is different.

Am I missing something here? From what I know, hopf map is from $S^{3}$ to $S^2$ but $S^{2}$ can be seen as $\mathbb{C}\cup\{\infty\}$ through stereo projection. Or is it that my interpretation of $h(z_{1},z_{2})=\frac{z_{1}}{z_{2}}$ is wrong.

Thanks.

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The Hopf map is really only defined up to some choice of cosmetic tweaking. Given any two rotations $R$ and $S$ of $\mathbb{R}^4$ and $\mathbb{R}^3$, you can swap one Hopf map $h:S^3\to S^2$ with another one given by the composition $S\circ h\circ R^{-1}$. In particular, the relationship between your two Hopf maps is that $c$ and $d$ are switched, which amounts to an improper reflection in the domain $S^3$. Mathworld has yet a different incarnation of the formula with coordinates permuted.

(Also one can perform a conformal automorphism of the Riemann sphere $\mathbb{C}\cup\{\infty\}$ instead of a rotation of the sphere $S^2$, since they amount to the same thing.)

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