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

{(X_1,X_2,X_3,X_4):X_1^2+X_2^2+X_3^2+X_4^2=1}, " src="https://mathworld.wolfram.com/images/equations/HopfMap/NumberedEquation1.gif" style="height:21px; width:284px" />

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and the 2-sphere is a submanifold of ,

{(x_1,x_2,x_3):x_1^2+x_2^2+x_3^2=1}. " src="https://mathworld.wolfram.com/images/equations/HopfMap/NumberedEquation2.gif" style="height:21px; width:211px" />

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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 .