Hopf bundle
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In mathematics, the Hopf bundle (or Hopf fibration), named after Heinz Hopf, is an important example of a fiber bundle. It has base space S2, total space S3, and fiber S1:
It was discovered by Heinz Hopf in 1931. The Hopf bundle also gives an example of a principal bundle by identifying the fiber with the circle group.
Key-Ring Model of the Hopf Fibration.To construct the Hopf bundle, consider S3 as the subset of all (z0, z1) in C2 such that |z0|2 + |z1|2 = 1. Identify (z0, z1) with (Î»z0, Î»z1) where Î» is a complex number with norm one. Then the quotient of S3 by this equivalence relation is the complex projective line, CP1, also known as the Riemann sphere S2. Clearly the fiber of a point is S1, and it is easy to show that local triviality holds, so that the Hopf bundle is a fiber bundle. The key-ring model in the picture can be mathematically described as a stereographic projection of S3 into R3. It does not show all the circles, of course (they would fill all of R3) but rather only those lying on a common torus in S3
Another way to look at the Hopf bundle is to regard S3 as the special unitary group SU(2). The group SU(2) is isomorphic to Spin(3) and so acts transitively on S2 by rotations. The stabilizer of a point is isomorphic to the circle group U(1). According to standard Lie group theory, SU(2) is then a principal U(1)-bundle over the left coset space SU(2)/U(1) which is diffeomorphic to the 2-sphere. The fibers in this bundle are just the left cosets of U(1) in SU(2).
In quantum mechanics, the Riemann sphere is known as the Bloch sphere, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system or qubit. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration .
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Hopf map
The Hopf map p : S3 â S2 is defined by
p (z0, z1) = (|z0|2 - |z1|2, 2z0z1*)
the first component is a real number, the second complex, so together they define a point in R3. It's easy to check that if |z0|2 + |z1|2 = 1, then p (z0, z1) lies on the unit 2-sphere. Conversely, if p (z0, z1) = p (z2, z3) then (z2, z3) = (Î»z0, Î»z1) for some unit Î».
Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic, but is of infinite order in Ï3(S2). In fact, the Hopf map generates Ï3(S2).
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Generalizations
More generally, the Hopf construction gives circle bundles p : S2n+1 â CPn over complex projective space. This is actually the restriction of the tautological line bundle over CPn to the unit sphere in Cn+1.
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Real, quaternionic, and octonionic Hopf bundles
One may also regard S1 as lying in R2 and factor out by unit real multiplication to obtain RP1 = S1 and a fiber bundle S1 â S1 with fiber S0. Similarly, one can regard S4nâ1 as lying in Hn (quaternionic n-space) and factor out by unit quaternion (= S3) multiplication to get HPn. In particular, since S4 = HP1, there is a bundle S7 â S4 with fiber S3. A similar construction with the octonions yields a bundle S15 â S8 with fiber S7. These bundles are sometimes also called Hopf bundles. As a consequence of Adams' theorem, these are the only fiber bundles with spheres as total space, base space, and fiber.
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