Hopf fibration

Example

Let $S^3 = \set{(z_0, z_1) \in \C^2 : \abs{z_0}^2 + \abs{z_1}^2 = 1}$, and identify $\R^3$ with $\C \times \R$. Define $h \colon S^3 \to \R^3$ by $$h(z_0, z_1) = \bigl(2 z_0 \overline{z_1}, \ \abs{z_0}^2 - \abs{z_1}^2\bigr).$$ The image lies in the unit sphere $S^2$, because $$\abs{2 z_0 \overline{z_1}}^2 + \bigl(\abs{z_0}^2 - \abs{z_1}^2\bigr)^2 = \bigl(\abs{z_0}^2 + \abs{z_1}^2\bigr)^2 = 1.$$ Two points of $S^3$ have the same image if and only if one is $(\lambda z_0, \lambda z_1)$ for the other $(z_0, z_1)$ and some $\lambda \in \C$ with $\abs{\lambda} = 1$. So each fibre of $h \colon S^3 \to S^2$ is a circle. The map $h$ is a fibre bundle with fibre $S^1$, hence a Serre fibration. Equivalently, $h$ is the quotient map $S^3 \to \C P^1 \cong S^2$, $(z_0, z_1) \mapsto [z_0 : z_1]$.