Hopf Map
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المرجع الالكتروني للمعلوماتيه
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www.almerja.com
الجزء والصفحة:
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13-5-2021
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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
![S^3=<span style=]() {(X_1,X_2,X_3,X_4):X_1^2+X_2^2+X_3^2+X_4^2=1}, " src="https://mathworld.wolfram.com/images/equations/HopfMap/NumberedEquation1.gif" style="height:21px; width:284px" /> |
(1)
|
and the 2-sphere is a submanifold of 
,
![S^2=<span style=]() {(x_1,x_2,x_3):x_1^2+x_2^2+x_3^2=1}. " src="https://mathworld.wolfram.com/images/equations/HopfMap/NumberedEquation2.gif" style="height:21px; width:211px" /> |
(2)
|
The Hopf map takes points (
, 
, 
, 
) on a 3-sphere to points on a 2-sphere (
, 
, 
)
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 
.
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