The Eilenberg-MacLane space is the classifying space for circle group principal bundles. By its very nature, it has a single nontrivial homotopy group, the second, and this is isomorphic to the group of integers
This means that there is, up to homotopy, a canonical (up to sign), continuous map from the 2-sphere
such that .
In other words, the Hopf fibration is the -bundle with unit first Chern class on .
An explicit topological space presenting the Hopf fibration may be obtained as follows.
Then the continuous function defined by
gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical line bundle.) Alternatively, if we use
and identify this presentation of the 2-sphere with the complex projective line via stereographic projection, the Hopf fibration is identified with the map given by sending
See Hopf construction.
For each of the normed division algebras over ,
there is a corresponding Hopf fibration of Hopf invariant one. The total space of the fibration is the space of pairs of unit norm: . These gives spheres of dimension 1, 3, 7, and 15 respectively. The base space of the fibration is projective 1-space , giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is a map
() which sends the pair to .
When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”. We may think of the homotopically as being the 3-dimensional Cartesian space with origin removed and think of this as being 3-dimensional physical space with a unit point magnetic charge at the origin removed. The corresponding electromagnetic field away from the origin is given by a connection on the corresponding Hopf fibration bundle.
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for to the topos of sheaves over .)
The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory? : they can be used to provide generators for the non-zero homotopy groups for the classifying space of the stable orthogonal group, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra ). [To be followed up on.]