Hopf bundle

In mathematics, the Hopf bundle (or Hopf Fibration) is a particular fiber bundle with base space sphere, total space S 3, and fiber unit circle: : S 1 → S 3 → S 2 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 S 3 to lie in C2. Identify ( z 0, z 1) with (λ z 0, λ z 1) where λ is a complex number with norm one. Then the quotient space of S 3 by this equivalence relation is the Riemann sphere S 2 also known as the complex projective line, CP1. Clearly the fiber of a point is S 1, 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 S 3 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 : S 3 → S 2 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 : S 2 n +1 → CP n over complex projective space. This is actually the restriction of the tautological line bundle over CP n to the unit sphere in C n +1.

==Real, quaternionic, and octonionic Hopf bundles==

One may also regard S 1 as lying in R2 and factor out by unit real multiplication to obtain RP1 = S 1 and a fiber bundle S 1 → S 1 with fiber S 0. Similarly, one can regard S 4 n −1 as lying in H n (quaternion n -space) and factor out by unit quaternion (= S 3) multiplication to get HP n . In particular, since S 4 = HP1, there is a bundle S 7 → S 4 with fiber S 3. A similar construction with the octonions yields a bundle S 15 → S 8 with fiber S 7. 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.