Hopf fibration




Homotopy theory

Manifolds and cobordisms



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

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(,2)BS 1K(\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

π 2(K(,2)). \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

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

such that [ϕ]π 2(K(,2))=±1[\phi] \in \pi_2(K(\mathbb{Z},2)) = \pm 1 \in \mathbb{Z}.

As any map into K(,2)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

S 1 S 3 S 2 ϕ BS 1K(,2). \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)U(1)-bundle with unit first Chern class on S 2S^2.

Realization via the complex numbers

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

S 3{(z 0,z 1)×||z 0| 2+|z 1| 2=1} S^3 \simeq \{(z_0, z_1) \in \mathbb{C}\times \mathbb{C} \,|\, {|z_0|}^2 + {|z_1|}^2 = 1\}


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

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

(z 0,z 1)z 0z 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{(z,x)×||z| 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 3S 2S^3 \to S^2 given by sending

(z 0,z 1)(2z 0z 1 *,|z 0| 2|z 1| 2). (z_0, z_1) \mapsto (2 z_0 z_1^* , {|z_0|}^2 - {|z_1|}^2).

Realization via the Hopf construction

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

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=,,,𝕆,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 (α,β)A 2(\alpha, \beta) \in A^2 of unit norm: |α| 2+|β| 2=1{|\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 1(A)\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 n1S 2 n1S^{2^n - 1} \to S^{2^{n-1}}

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

Via the Hopf construction

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

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


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 2S^2 homotopically as being the 3-dimensional Cartesian space with origin removed 3{0}\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 HH which generates the cohomology ring K U(S 2)K_U(S^2), and satisfying the relation H 2=2H1H^2 = 2 \cdot H - 1, or (H1) 2=0(H-1)^2 = 0. (So in particular HH has an inverse H 1=2HH^{-1} = 2- H , see at Bott generator.)

A succinct formulation of Bott periodicity for complex K-theory is that for a space XX whose homotopy type is that of a CW-complex, we have

K(S 2×X)K(S 2)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)K(S^2) to the topos of sheaves over XX.)

The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory K OK_O: they can be used to provide generators for the non-zero homotopy groups π n(BO)\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 𝕆\mathbb{O}). [To be followed up on.]


Reviews include

Revised on December 1, 2015 03:51:33 by David Corfield (