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  arises in many contexts, and can be generalized to a map . For any point  in the sphere, its preimage  is a circle  in . There are several descriptions of the Hopf map, also called the Hopf fibration.

As a submanifold of , the 3-sphere is

(1)

and the 2-sphere is a submanifold of ,

(2)

The Hopf map takes points (, , , ) on a 3-sphere to points on a 2-sphere (, , )

			(3)
			(4)
			(5)

Every point on the 2-sphere corresponds to a circle called the Hopf circle on the 3-sphere.

By stereographic projection, the 3-sphere can be mapped to , where the point at infinity corresponds to the north pole. As a map, from , the Hopf map can be pretty complicated. The diagram above shows some of the preimages , called Hopf circles. The straight red line is the circle through infinity.

By associating  with , the map is given by , which gives the map to the Riemann sphere.

The Hopf fibration is a fibration

(6)

and is in fact a principal bundle. The associated vector bundle

(7)

where

(8)

is a complex line bundle on . In fact, the set of line bundles on the sphere forms a group under vector bundle tensor product, and the bundle  generates all of them. That is, every line bundle on the sphere is  for some .

The sphere  is the Lie group of unit quaternions, and can be identified with the special unitary group , which is the simply connected double cover of . The Hopf bundle is the quotient map .