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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 f:S^3->S^2 arises in many contexts, and can be generalized to a map S^7->S^4. For any point p in the sphere, its preimage f^(-1)(p) is a circle S^1 in S^3. There are several descriptions of the Hopf map, also called the Hopf fibration.

As a submanifold of R^4, the 3-sphere is

 S^3={(X_1,X_2,X_3,X_4):X_1^2+X_2^2+X_3^2+X_4^2=1},
(1)

and the 2-sphere is a submanifold of R^3,

 S^2={(x_1,x_2,x_3):x_1^2+x_2^2+x_3^2=1}.
(2)

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)

x_1=2(X_1X_2+X_3X_4)
(3)
x_2=2(X_1X_4-X_2X_3)
(4)
x_3=(X_1^2+X_3^2)-(X_2^2+X_4^2).
(5)

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

HopfMap

By stereographic projection, the 3-sphere can be mapped to R^3, where the point at infinity corresponds to the north pole. As a map, from R^3, the Hopf map can be pretty complicated. The diagram above shows some of the preimages f^(-1)(p), called Hopf circles. The straight red line is the circle through infinity.

By associating R^4 with C^2, the map is given by f(z,w)=z/w, which gives the map to the Riemann sphere.

The Hopf fibration is a fibration

 S^1->S^3->S^2,
(6)

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

 L=S^3×C/U(1),
(7)

where

 ((z,w),v)∼((e^(it)z,e^(it)w),e^(-it)v)
(8)

is a complex line bundle on S^2. In fact, the set of line bundles on the sphere forms a group under vector bundle tensor product, and the bundle L generates all of them. That is, every line bundle on the sphere is L^( tensor k) for some k.

The sphere S^3 is the Lie group of unit quaternions, and can be identified with the special unitary group SU(2), which is the simply connected double cover of SO(3). The Hopf bundle is the quotient map S^2=SU(2)/U(1).


See also

Fibration, Fiber Bundle, Homogeneous Space, Principal Bundle, Stereographic Projection, Vector Bundle

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Hopf Map." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HopfMap.html

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