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.