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with base space S2, total space S3, and fiber S1. 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.
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).
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.
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 ( quaternionicQuaternion is also a musical composition by Sofia Gubaidulina In mathematics, the quaternions are a non-commutative extension of the complex numbers. They were first described by William Rowan Hamilton of Ireland in 1843. At first, the quaternions were re 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 octonionIn mathematics, the octonions are a nonassociative extension of the quaternions. They form an 8-dimensional normed division algebra over the real numbers. The octonion algebra is often denoted O . Lacking the desirable property of associativity, the octons 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.
Algebraic topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca Differential geometry Bundles (mathematics) Homotopy theory