The Hopf map is the first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic (Hopf 1931). 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 line 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 
.
