The first example discovered of a map from a higher-dimensional sphere to a lower-dimensional sphere which is not null-homotopic. Its discovery was a shock to the mathematical community, since it was believed at the time that all such maps were null-homotopic, by analogy with homology groups.
The Hopf map arises in many contexts, and can be generalized to a map . For any point in the sphere, its preimage is a circle in . There are several descriptions of the Hopf map, also called the Hopf fibration.
As a submanifold of , the 3-sphere is
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and the 2-sphere is a submanifold of ,
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The Hopf map takes points (, , , ) on a 3-sphere to points on a 2-sphere (, , )
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Every point on the 2-sphere corresponds to a circle called the Hopf circle on the 3-sphere.
By stereographic projection, the 3-sphere can be mapped to , where the point at infinity corresponds to the north pole. As a map, from , the Hopf map can be pretty complicated. The diagram above shows some of the preimages , called Hopf circles. The straight red line is the circle through infinity.
By associating with , the map is given by , which gives the map to the Riemann sphere.
The Hopf fibration is a fibration
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and is in fact a principal bundle. The associated vector bundle
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where
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is a complex line bundle on . In fact, the set of line bundles on the sphere forms a group under vector bundle tensor product, and the bundle generates all of them. That is, every line bundle on the sphere is for some .
The sphere is the Lie group of unit quaternions, and can be identified with the special unitary group , which is the simply connected double cover of . The Hopf bundle is the quotient map .