In the mathematical field of differential topology, the Hopf bundle (also known as the Hopf fibration or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle, which is important within twistor theory.

Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct great circle of the 3-sphere (Hopf 1931). [1] Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.

This fiber bundle structure is denoted $$ S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}} \, S^{2}, $$ meaning that the fiber space $ S^1 $ (a circle) is embedded in the total space $S^3$ (the 3-sphere), and $ p : S^3 \rightarrow S^2 $ (Hopf's map) projects $S^3$ onto the base space $S^2$ (the ordinary 2-sphere). The Hopf bundle, like any fiber bundle, has the important property that it is locally a product space. However it is not a trivial fiber bundle, i.e., $ S^3 $ is not globally a product of $ S^2 $ and $ S^1 $ although locally it is indistinguishable from it.

This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group.

Stereographic projection of the Hopf bundle induces a remarkable structure on $ \mathbb{R}^3, $ in which all of 3-dimensional space, except for the $z$-axis, is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the Stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.)

These tori are illustrated in the images at right. [Move these images to a KMR page] When $ \mathbb{R}^3 $ is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see Topology and geometry). The loops are homeomorphic to circles, although they are not geometric circles.

There are numerous generalizations of the Hopf bundle. The unit sphere in complex coordinate space $ \mathbb{C}^{n+1} $ fibers naturally over the complex projective space $ \mathbb{C}{\mathbb{P}}^n $ with circles as fibers, and there are also real, quaternionic, [2] and octonionic versions of these fiber bundles. In particular, the Hopf bundle belongs to a family of four fiber bundles in which the total space, the base space, and the fiber space are all spheres: $$ S^{0} \hookrightarrow S^{1} \to S^{1}, $$ $$ S^{1} \hookrightarrow S^{3} \to S^{2}, $$ $$ S^{3} \hookrightarrow S^{7} \to S^{4}, $$ $$ S^{7} \hookrightarrow S^{15} \to S^{8}. $$

By Adams's theorem such fiber bundles can occur only in these dimensions.

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• The Hopf bundle visualized by Carlos Farias.
• Blackboard explanation of the Hopf bundle by Niles Johnson.
• Visualizations of the Hopf bundle by Niles Johnson.
• More on the Hopf bundle by Niles Johnson.
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• Group action
• Klein geometry
• Introduction to Gauge theory
• Gauge theory (physics)
• Gauge theory (mathematics)
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Contents:
• 1 Definition and construction
• 1.1 Direct construction
• 1.1.1 Geometric interpretation using the complex projective line
• 1.1.2 Fiber bundle structure
• 1.2 Geometric interpretation using rotations
• 1.2.1 Explicit formulae
• 1.3 Fluid mechanics
• 2 Generalizations
• 2.1 Real Hopf fibrations
• 2.2 Complex Hopf fibrations
• 2.3 Quaternionic Hopf fibrations
• 2.4 Octonionic Hopf fibrations
• 2.5 Fibrations between spheres
• 3 Geometry and applications
• 4 Notes
• 5 References
• 6 External links

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