CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
The Hopf fibration (named after Heinz Hopf) is a canonical nontrivial circle bundle over the 2-sphere whose total space is the 3-sphere.
The Eilenberg-MacLane space $K(\mathbb{Z},2) \simeq B S^1$ 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 $[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}$.
As any map into $K(\mathbb{Z},2)$ this classifies a circle group principal bundle over its domain. This is the Hopf fibration, fitting into the long fiber sequence
In other words, the Hopf fibration is the $U(1)$-bundle with unit first Chern class on $S^2$.
An explicit topological space presenting the Hopf fibration may be obtained as follows.
Identify
and
Then the continuous function $S^3 \to S^2$ 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 $S^3 \to S^2$ given by sending
See Hopf construction.
For each of the normed division algebras over $\mathbb{R}$,
there is a corresponding Hopf fibration of Hopf invariant one. The total space of the fibration is the space of pairs $(\alpha, \beta) \in A^2$ of unit norm: ${|\alpha|}^2 + {|\beta|}^2 = 1$. These gives spheres of dimension 1, 3, 7, and 15 respectively. The base space of the fibration is projective 1-space $\mathbb{P}^1(A)$, giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is a map
($n = 1, 2, 3, 4$) which sends the pair $(\alpha, \beta)$ to $\alpha/\beta$.
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 $S^2$ homotopically as being the 3-dimensional Cartesian space with origin removed $\mathbb{R}^3 - \{0\}$ 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.
In complex K-theory, the Hopf fibration represents a class $H$ which generates the cohomology ring $K_U(S^2)$, and satisfying the relation $H^2 = 2 \cdot H - 1$, or $(H-1)^2 = 0$. (So in particular $H$ has an inverse $H^{-1} = 2- H$, see at Bott generator.)
A succinct formulation of Bott periodicity for complex K-theory is that for a space $X$ whose homotopy type is that of a CW-complex, we have
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for $K(S^2)$ to the topos of sheaves over $X$.)
The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory? $K_O$: they can be used to provide generators for the non-zero homotopy groups $\pi_n(B O)$ 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 $\mathbb{O}$). [To be followed up on.]