Contents

Contents

Idea

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

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

Its canonically associated complex line bundle is the basic line bundle on the 2-sphere.

This we discuss below in

More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in

On the 3-sphere

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

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

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

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

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

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

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

Realization via the complex numbers

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

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

and

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

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

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

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

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

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

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

Realization via quaternions

Alternatively, we may regard $S^3 \simeq S(\mathbb{H})$ as the unit sphere in the quaternions and $S^2 \simeq S\left( \mathbb{H}_{\mathrm{im}}\right)$ as the unit sphere in the imaginary quaternions. Under this identification, the complex Hopf fibration is equivalently represented by

$\array{ S(\mathbb{H}) &\longrightarrow& S\left( \mathbb{H}_{\mathrm{im}}\right) \\ q &\mapsto& q \cdot \mathbf{i} \cdot \overline{q} }$

where $\mathbf{i} \in S\left( \mathbb{H}_{\mathrm{im}}\right)$ is any unit imaginary quaternion.

Realization via the Hopf construction

Regard $S^1 = U(1)$ as equipped with its circle group structure. This makes $S^1$ in particular an H-space. The Hopf fibration $S^1 \to S^3 \to S^2$ is the Hopf construction applied to this H-space.

Spin(3)-equivariance

Consider

1. the Spin(3)-action on the 2-sphere $S^2$ which is induced by the defining action on $\mathbb{R}^3$ under the identification $S^2 \simeq S(\mathbb{R}^3)$;

2. the Spin(3)-action on the 3-sphere $S^3$ which is induced under the exceptional isomorphism $Spin(3) \simeq Sp(1) = U(1,\mathbb{H})$ by the canonical left action of $U(1,\mathbb{H})$ on $\mathbb{H}$ via $S^3 \simeq S(\mathbb{H})$.

Then the complex Hopf fibration $S^3 \overset{h_{\mathbb{C}}}{\longrightarrow} S^2$ is equivariant with respect to these actions.

A way to make the $Spin(3)$-equivariance of the complex Hopf fibration fully explicit is to observe that it is equivalently the following map of coset spaces:

$\array{ S^1 &\overset{fib(h_{\mathbb{C}})}{\longrightarrow}& S^{3} &\overset{h_{\mathbb{C}}}{\longrightarrow}& S^2 \\ = && = && = \\ \frac{Spin(2)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(1)} &\longrightarrow& \frac{Spin(3)}{Spin(2)} }$

On the 1-sphere, 3-sphere, 7-sphere and 15-sphere

Via norms and projections

For each of the normed division algebras over $\mathbb{R}$, the real numbers, complex numbers, quaternions, octonions

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

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: ${|\alpha|}^2 + {|\beta|}^2 = 1$. This 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 the map

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

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

Via the Hopf construction

When $X$ is a sphere that is an $H$-space, namely, one of the groups $S^0 = 1$ the trivial group, $S^1 = \mathbb{Z}/2$ the group of order 2, the 3-sphere special unitary group $S^3 = SU(2)$; or the 7-sphere $S^7$ with its Moufang loop structure, then the Hopf construction produces the above four Hopf fibrations:

1. $S^0 \hookrightarrow S^1 \to S^1$real Hopf fibration
2. $S^1 \hookrightarrow S^3 \to S^2$complex Hopf fibration
3. $S^3 \hookrightarrow S^7 \to S^4$quaternionic Hopf fibration
4. $S^7 \hookrightarrow S^{15} \to S^8$octonionic Hopf fibration

Properties

Relation to stable homotopy groups of spheres

Let

$H_{\mathbb{C}} \in \pi_3(S^2)$,

$H_{\mathbb{H}} \in \pi_7(S^4)$

$H_{\mathbb{O}} \in \pi_{15}(S^8)$

be the homotopy class of the complex Hopf fibration, the quaternionic Hopf fibration and the octonionic Hopf fibration, respectively. Then their suspensions are the generators of the corresponding stable homotopy groups of spheres:

\begin{aligned} \Sigma H_{\mathbb{C}} & = \pm 1 \in \mathbb{Z}_2 \simeq \pi_1^{st} \\ \Sigma H_{\mathbb{H}} & = \pm 1 \in \mathbb{Z}_{24} \simeq \pi_3^{st} \\ \Sigma H_{\mathbb{O}} & = \pm 1 \in \mathbb{Z}_{240} \simeq \pi_7^{st} \end{aligned}

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 $\mathbb{R}^3 - \{0\}$ 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.

K-theory

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

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

(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 $\pi_n(B O)$ 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.]

References

Exposition:

• Saifuddin Syed, Group structure on spheres and the Hopf fibration (pdf)

Review:

Formulation in homotopy type theory:

Relation to skyrmions:

Discussion of supersymmetric Hopf fibrations:

Last revised on November 27, 2020 at 08:42:10. See the history of this page for a list of all contributions to it.