Hopf bundle
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In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space S2, total space S3, and fiber S1:
- S1 → S3 → S2
It was discovered by Heinz Hopf in 1931. The Hopf bundle can actually be considered as a principal bundle when the fiber is identified with the circle group.
To construct the Hopf bundle, consider S3 to lie in C2. Identify (z0, z1) with (λz0, λz1) where λ is a complex number with norm one. Then the quotient of S3 by this equivalence relation is the Riemann sphere S2 also known as the complex projective line, CP1. 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.
- [a picture of the Hopf bundle would be nice here]
Another way to look at the Hopf bundle is to regard S3 as the special unitary group SU(2). The diagonal subgroup of SU(2) is isomorphic to the circle group U(1). This is a closed Lie subgroup of SU(2). According to standard Lie group theory, SU(2) is then a principal U(1)-bundle over the left coset space SU(2)/U(1). One can show that SU(2)/U(1) is diffeomorphic to the 2-sphere. The fibers in this bundle are just the left cosets of U(1) in SU(2).
Hopf proved that the Hopf map p : S3 → S2 has Hopf invariant 1, and therefore is not null-homotopic, but is of infinite order in π3(S2). In fact, the Hopf map generates π3(S2).
Generalizations
More generally, the Hopf construction gives circle bundles p : S2n+1 → CPn over complex projective space. This is actually the restriction of the tautological line bundle over CPn to the unit sphere in Cn+1.
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 → S1 with fiber S0. Similarly, one can regard S4n−1 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 → S4 with fiber S3. A similar construction with the octonions yields a bundle S15 → 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.