nLab
Hopf fibration

Idea

The Hopf fibration (named after Heinz Hopf) is a canonical nontrivial circle bundle over the 2-sphere whose total space is the 3-sphere.

$$S^1\hookrightarrow S^3\to S^2$$S^1 \hookrightarrow S^3 \to S^2

Definition

Homotopy-theoretic characterization

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

$${\pi }_2(K(\mathbb{Z},2))\simeq \mathbb{Z}.$$\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z} \,.

This means that there is, up to homotopy, a canonical (up to sign), continuous map from the 2-sphere

$$\varphi :S^2\to K(\mathbb{Z},2),$$\phi : S^2 \to K(\mathbb{Z},2) \,,

such that $[\varphi ]\in {\pi }_2(K(\mathbb{Z},2))=\pm 1\in \mathbb{Z}$.

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

$$\begin{array}{ccc}{S}^1& \hookrightarrow & S^3\\ & & \downarrow \\ & & S^2& \stackrel{\varphi }{\to }& B{S}^1\simeq K(\mathbb{Z},2)\end{array}.$$\array{ S^1 &\hookrightarrow& S^3 \\ && \downarrow \\ && S^2 &\stackrel{\phi}{\to}& B S^1 \simeq K(\mathbb{Z},2) } \,.

In other words, the Hopf fibration is the $U(1)$-bundle with unit first Chern class on $S^2$.

Explicit model

An explicit topological space presenting the Hopf fibration may be obtained as follows.

Identify

$$S^3\simeq \{(z_0,z_1)\in ℂ\times ℂ\mid {\mid z_0\mid }^2+{\mid z_1\mid }^2=1\}$$S^3 \simeq \{(z_0, z_1) \in \mathbb{C}\times \mathbb{C} \,|\, {|z_0|}^2 + {|z_1|}^2 = 1\}

and

$$S^2\simeq {\mathrm{ℂ\mathbb{P}}}^1\simeq ℂ\bigsqcup \{\mathrm{\infty }\}$$S^2 \simeq \mathbb{C P}^1 \simeq \mathbb{C} \sqcup \{\infty\}

Then the continuous function $S^3\to S^2$ defined by

$$(z_0,z_1)\mapsto \frac{z_0}{z_1}$$(z_0, z_1) \mapsto \frac{z_0}{z_1}

gives the Hopf fibration. (Thus, the Hopf fibration is a circle bundle naturally associated with the canonical line bundle.) Alternatively, if we use

$$S^2\simeq \{(z,x)\in ℂ\times ℝ\mid {\mid z\mid }^2+x^2=1\}.$$S^2 \simeq \{(z, x) \in \mathbb{C} \times \mathbb{R} \,|\, {|z|}^2 + x^2 = 1\} \,.

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

$$(z_0,z_1)\mapsto (2z_0{z}_1^{*},{\mid z_0\mid }^2-{\mid z_1\mid }^2).$$(z_0, z_1) \mapsto (2 z_0 z_1^* , {|z_0|}^2 - {|z_1|}^2).

Homotopy-theoretic construction

See Hopf construction.

Variations

For each of the normed division algebras over $ℝ$,

$$A=ℝ,ℂ,ℍ,𝕆,$$A = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O},

there is a corresponding Hopf fibration. The total space of the fibration is the space of pairs $(\alpha ,\beta )\in A^2$ of unit norm: ${\mid \alpha \mid }^2+{\mid \beta \mid }^2=1$. 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

$$S^{2^n-1}\to S^{2^{n-1}}$$S^{2^n - 1} \to S^{2^{n-1}}

($n=1,2,3,4$) which sends the pair $(\alpha ,\beta )$ to $\alpha /\beta$.

Applications

Magnetic monopoles

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 ${ℝ}^3-\{0\}$ 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.

K-theory

In complex K-theory, the Hopf fibration represents a class $H$ which generates the ring $K_U(S^2)$, and satisfying the relation $H^2=2\cdot H-1$, or $(H-1{)}^2=0$. A succinct formulation of Bott periodicity? for complex K-theory is that for a space $X$ whose homotopy type is that if a CW-complex, we have

$$K(S^2\times X)\cong K(S^2)\otimes K(X)$$K(S^2 \times X) \cong K(S^2) \otimes K(X)

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 ${\pi }_n(BO)$, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra $𝕆$). To be followed up on.