Hopf Map

The first example discovered of a Map from a higher-dimensional Sphere to a lower-dimensional Sphere which is not null-Homotopic. Its discovery was a shock to the mathematical community, since it was believed at the time that all such maps were null-Homotopic, by analogy with Homology Groups. The Hopf map takes points ($X_1$, $X_2$, $X_3$, $X_4$) on a 3-sphere to points on a 2-sphere ($x_1$, $x_2$, $x_3$)

$$\begin{eqnarray*} x_1&=&2(X_1X_2+X_3X_4)\\ x_2&=&2(X_1X_4-X_2X_3)\\ x_3&=&({X_1}^2+{X_3}^2)-({X_2}^2+{X_4}^2). \end{eqnarray*}$$

Every point on the two Spheres corresponds to a Circle called the Hopf Circle on the 3-Sphere.