Hopf fibration

In <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a <a href="/facts/3-sphere/u67nImpY">3-sphere</a> (a <a href="/facts/Hypersphere/bFu2S7E2">hypersphere</a> in <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional space</a>) in terms of <a href="/facts/Circles/9fy5ggKn">circles</a> and an ordinary <a href="/facts/Sphere/jywYLNM2">sphere</a>. Discovered by <a href="/facts/Heinz_Hopf/PMks10NV">Heinz Hopf</a> in 1931, it is an influential early example of a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a>. Technically, Hopf found a many-to-one <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere is mapped from a distinct <a href="/facts/Great_circle/x4fq6eNm">great circle</a> of the 3-sphere (Hopf 1931). 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

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    {\displaystyle S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}}S^{2},}

meaning that the fiber space S1 (a circle) is <a href="/facts/Embedding/HKeVKc42">embedded</a> in the total space S3 (the 3-sphere), and p : S3 → S2 (Hopf's map) projects S3 onto the base space S2 (the ordinary 2-sphere). The Hopf fibration, like any fiber bundle, has the important property that it is <a href="/facts/Locally_trivial/SJ7Ek6cI">locally</a> a <a href="/facts/Product_space/0hg02IaX">product space</a>. However it is not a trivial fiber bundle, i.e., S3 is not globally a product of S2 and S1 although locally it is indistinguishable from it.
This has many implications: for example the existence of this bundle shows that the higher <a href="/facts/Homotopy_groups_of_spheres/YMENlGgd">homotopy groups of spheres</a> are not trivial in general. It also provides a basic example of a <a href="/facts/Principal_bundle/bbB2D94B">principal bundle</a>, by identifying the fiber with the <a href="/facts/Circle_group/kuzdWtHo">circle group</a>.
<a href="/facts/Stereographic_projection/oVyMrWqR">Stereographic projection</a> of the Hopf fibration induces a remarkable structure on R3, in which all of 3-dimensional space, except for the z-axis, is filled with nested <a href="/facts/Torus/J9SkaV1W">tori</a> made of linking <a href="/facts/Villarceau_circles/rEjnEru8">Villarceau circles</a>. Here each fiber projects to a <a href="/facts/Circle/9fy5ggKn">circle</a> in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the stereographic projection of the <a href="/facts/Inverse_image/KANefMAl">inverse image</a> 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. When R3 is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see <a href="/facts/Geometry/wTQf5ffQ">Topology and geometry</a>). The loops are <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to circles, although they are not geometric <a href="/facts/Circle/9fy5ggKn">circles</a>.
There are numerous generalizations of the Hopf fibration. The unit sphere in <a href="/facts/Complex_coordinate_space/4zCD19qb">complex coordinate space</a> Cn+1 fibers naturally over the <a href="/facts/Complex_projective_space/QmgneIzS">complex projective space</a> CPn with circles as fibers, and there are also <a href="/facts/Real_number/R02gw5Pb">real</a>, <a href="/facts/Quaternion/T3tnu7mX">quaternionic</a>, and <a href="/facts/Octonion/nC2sIz3p">octonionic</a> versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres:

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    {\displaystyle S^{0}\hookrightarrow S^{1}\to S^{1},}

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    {\displaystyle S^{1}\hookrightarrow S^{3}\to S^{2},}

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    {\displaystyle S^{3}\hookrightarrow S^{7}\to S^{4},}

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    {\displaystyle S^{7}\hookrightarrow S^{15}\to S^{8}.}

By <a href="/facts/Hopf_invariant/6MCJDeDC"> Adams's theorem</a> such fibrations can occur only in these dimensions.

Hopf fibration open-in-new

Hopf fibration