## 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 (, , , ) on a 3-sphere to points on a 2-sphere (, , )

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