Hopf fibration

Identify R⁴ with C² and R³ with C × R (where C denotes the complex numbers) by writing:

${\displaystyle (x_{1},x_{2},x_{3},x_{4})\leftrightarrow (z_{0},z_{1})=(x_{1}+ix_{2},x_{3}+ix_{4})}$

and

${\displaystyle (x_{1},x_{2},x_{3})\leftrightarrow (z,x)=(x_{1}+ix_{2},x_{3})}$.

Thus S³ is identified with the subset of all (z₀, z₁) in C² such that |z₀|² + |z₁|² = 1, and S² is identified with the subset of all (z, x) in C×R such that |z|² + x² = 1. (Here, for a complex number z = x + iy, |z|² = z z^∗ = x² + y², where the star denotes the complex conjugate.) Then the Hopf fibration p is defined by

${\displaystyle p(z_{0},z_{1})=(2z_{0}z_{1}^{\ast },\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}).}$

The first component is a complex number, whereas the second component is real. Any point on the 3-sphere must have the property that |z₀|² + |z₁|² = 1. If that is so, then p(z₀, z₁) lies on the unit 2-sphere in C × R, as may be shown by adding the squares of the absolute values of the complex and real components of p

${\displaystyle 2z_{0}z_{1}^{\ast }\cdot 2z_{0}^{\ast }z_{1}+\left(\left|z_{0}\right|^{2}-\left|z_{1}\right|^{2}\right)^{2}=4\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{0}\right|^{4}-2\left|z_{0}\right|^{2}\left|z_{1}\right|^{2}+\left|z_{1}\right|^{4}=\left(\left|z_{0}\right|^{2}+\left|z_{1}\right|^{2}\right)^{2}=1}$

Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if p(z₀, z₁) = p(w₀, w₁), then (w₀, w₁) must equal (λ z₀, λ z₁) for some complex number λ with |λ|² = 1. The converse is also true; any two points on the 3-sphere that differ by a common complex factor λ map to the same point on the 2-sphere. These conclusions follow, because the complex factor λ cancels with its complex conjugate λ^∗ in both parts of p: in the complex 2z₀z₁^∗ component and in the real component |z₀|² − |z₁|².

Since the set of complex numbers λ with |λ|² = 1 form the unit circle in the complex plane, it follows that for each point m in S², the inverse image p⁻¹(m) is a circle, i.e., p⁻¹m ≅ S¹. Thus the 3-sphere is realized as a disjoint union of these circular fibers.

A direct parametrization of the 3-sphere employing the Hopf map is as follows.[3]

${\displaystyle z_{0}=e^{i\,{\frac {\xi _{1}+\xi _{2}}{2}}}\sin \eta }$

${\displaystyle z_{1}=e^{i\,{\frac {\xi _{2}-\xi _{1}}{2}}}\cos \eta .}$

or in Euclidean R⁴

${\displaystyle x_{1}=\cos \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }$

${\displaystyle x_{2}=\sin \left({\frac {\xi _{1}+\xi _{2}}{2}}\right)\sin \eta }$

${\displaystyle x_{3}=\cos \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }$

${\displaystyle x_{4}=\sin \left({\frac {\xi _{2}-\xi _{1}}{2}}\right)\cos \eta }$

Where η runs over the range from 0 to π/2, ξ₁ runs over the range from 0 to 2π, and ξ₂ can take any value from 0 to 4π. Every value of η, except 0 and π/2 which specify circles, specifies a separate flat torus in the 3-sphere, and one round trip (0 to 4π) of either ξ₁ or ξ₂ causes you to make one full circle of both limbs of the torus.

A mapping of the above parametrization to the 2-sphere is as follows, with points on the circles parametrized by ξ₂.

${\displaystyle z=\cos(2\eta )}$

${\displaystyle x=\sin(2\eta )\cos \xi _{1}}$

${\displaystyle y=\sin(2\eta )\sin \xi _{1}}$

A geometric interpretation of the fibration may be obtained using the complex projective line, CP¹, which is defined to be the set of all complex one-dimensional subspaces of C². Equivalently, CP¹ is the quotient of C²\{0} by the equivalence relation which identifies (z₀, z₁) with (λ z₀, λ z₁) for any nonzero complex number λ. On any complex line in C² there is a circle of unit norm, and so the restriction of the quotient map to the points of unit norm is a fibration of S³ over CP¹.

CP¹ is diffeomorphic to a 2-sphere: indeed it can be identified with the Riemann sphere C_∞ = C ∪ {∞}, which is the one point compactification of C (obtained by adding a point at infinity). The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space. Alternatively, the point (z₀, z₁) can be mapped to the ratio z₁/z₀ in the Riemann sphere C_∞.

The Hopf fibration defines a fiber bundle, with bundle projection p. This means that it has a "local product structure", in the sense that every point of the 2-sphere has some neighborhood U whose inverse image in the 3-sphere can be identified with the product of U and a circle: p⁻¹(U) ≅ U × S¹. Such a fibration is said to be locally trivial.

For the Hopf fibration, it is enough to remove a single point m from S² and the corresponding circle p⁻¹(m) from S³; thus one can take U = S²\{m}, and any point in S² has a neighborhood of this form.

Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the 2-sphere in ordinary 3-dimensional space. The rotation group SO(3) has a double cover, the spin group Spin(3), diffeomorphic to the 3-sphere. The spin group acts transitively on S² by rotations. The stabilizer of a point is isomorphic to the circle group; its elements are angles of rotation leaving the given point unmoved, all sharing the axis connecting that point to the sphere's center. It follows easily that the 3-sphere is a principal circle bundle over the 2-sphere, and this is the Hopf fibration.

To make this more explicit, there are two approaches: the group Spin(3) can either be identified with the group Sp(1) of unit quaternions, or with the special unitary group SU(2).

In the first approach, a vector (x₁, x₂, x₃, x₄) in R⁴ is interpreted as a quaternion q ∈ H by writing

${\displaystyle q=x_{1}+\mathbf {i} x_{2}+\mathbf {j} x_{3}+\mathbf {k} x_{4}.\,\!}$

The 3-sphere is then identified with the versors, the quaternions of unit norm, those q ∈ H for which |q|² = 1, where |q|² = q q^∗, which is equal to x₁² + x₂² + x₃² + x₄² for q as above.

On the other hand, a vector (y₁, y₂, y₃) in R³ can be interpreted as a pure quaternion

${\displaystyle p=\mathbf {i} y_{1}+\mathbf {j} y_{2}+\mathbf {k} y_{3}.\,\!}$

Then, as is well-known since Cayley (1845), the mapping

${\displaystyle p\mapsto qpq^{*}\,\!}$

is a rotation in R³: indeed it is clearly an isometry, since |q p q^∗|² = q p q^∗ q p^∗ q^∗ = q p p^∗ q^∗ = |p|², and it is not hard to check that it preserves orientation.

In fact, this identifies the group of versors with the group of rotations of R³, modulo the fact that the versors q and −q determine the same rotation. As noted above, the rotations act transitively on S², and the set of versors q which fix a given right versor p have the form q = u + v p, where u and v are real numbers with u² + v² = 1. This is a circle subgroup. For concreteness, one can take p = k, and then the Hopf fibration can be defined as the map sending a versor ω to ω k ω^∗. All the quaternions ωq, where q is one of the circle of versors that fix k, get mapped to the same thing (which happens to be one of the two 180° rotations rotating k to the same place as ω does).

Another way to look at this fibration is that every versor ω moves the plane spanned by {1, k} to a new plane spanned by {ω, ωk}. Any quaternion ωq, where q is one of the circle of versors that fix k, will have the same effect. We put all these into one fibre, and the fibres can be mapped one-to-one to the 2-sphere of 180° rotations which is the range of ωkω^*.

This approach is related to the direct construction by identifying a quaternion q = x₁ + i x₂ + j x₃ + k x₄ with the 2×2 matrix:

${\displaystyle {\begin{bmatrix}x_{1}+\mathbf {i} x_{2}&x_{3}+\mathbf {i} x_{4}\\-x_{3}+\mathbf {i} x_{4}&x_{1}-\mathbf {i} x_{2}\end{bmatrix}}.\,\!}$

This identifies the group of versors with SU(2), and the imaginary quaternions with the skew-hermitian 2×2 matrices (isomorphic to C × R).

The rotation induced by a unit quaternion q = w + i x + j y + k z is given explicitly by the orthogonal matrix

${\displaystyle {\begin{bmatrix}1-2(y^{2}+z^{2})&2(xy-wz)&2(xz+wy)\\2(xy+wz)&1-2(x^{2}+z^{2})&2(yz-wx)\\2(xz-wy)&2(yz+wx)&1-2(x^{2}+y^{2})\end{bmatrix}}.}$

Here we find an explicit real formula for the bundle projection by noting that the fixed unit vector along the z axis, (0,0,1), rotates to another unit vector,

${\displaystyle {\Big (}2(xz+wy),2(yz-wx),1-2(x^{2}+y^{2}){\Big )},\,\!}$

which is a continuous function of (w, x, y, z). That is, the image of q is the point on the 2-sphere where it sends the unit vector along the z axis. The fiber for a given point on S² consists of all those unit quaternions that send the unit vector there.

We can also write an explicit formula for the fiber over a point (a, b, c) in S². Multiplication of unit quaternions produces composition of rotations, and

${\displaystyle q_{\theta }=\cos \theta +\mathbf {k} \sin \theta }$

is a rotation by 2θ around the z axis. As θ varies, this sweeps out a great circle of S³, our prototypical fiber. So long as the base point, (a, b, c), is not the antipode, (0, 0, −1), the quaternion

${\displaystyle q_{(a,b,c)}={\frac {1}{\sqrt {2(1+c)}}}(1+c-\mathbf {i} b+\mathbf {j} a)}$

will send (0, 0, 1) to (a, b, c). Thus the fiber of (a, b, c) is given by quaternions of the form q_{(a, b, c)}q_θ, which are the S³ points

${\displaystyle {\frac {1}{\sqrt {2(1+c)}}}{\Big (}(1+c)\cos(\theta ),a\sin(\theta )-b\cos(\theta ),a\cos(\theta )+b\sin(\theta ),(1+c)\sin(\theta ){\Big )}.\,\!}$

Since multiplication by q_(a,b,c) acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle.

The final fiber, for (0, 0, −1), can be given by defining q_(0,0,−1) to equal i, producing

${\displaystyle {\Big (}0,\cos(\theta ),-\sin(\theta ),0{\Big )},}$

which completes the bundle. But note that this one-to-one mapping between S³ and S²×S¹ is not continuous on this circle, reflecting the fact that S³ is not topologically equivalent to S²×S¹.

Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the 3-sphere is equivalent to a quaternion, which in turn is equivalent to a particular rotation of a Cartesian coordinate frame in three dimensions. The set of all possible quaternions produces the set of all possible rotations, which moves the tip of one unit vector of such a coordinate frame (say, the z vector) to all possible points on a unit 2-sphere. However, fixing the tip of the z vector does not specify the rotation fully; a further rotation is possible about the z-axis. Thus, the 3-sphere is mapped onto the 2-sphere, plus a single rotation.

The rotation can be represented using the Euler angles θ, φ, and ψ. The Hopf mapping maps the rotation to the point on the 2-sphere given by θ and φ, and the associated circle is parametrized by ψ. Note that when θ = π the Euler angles φ and ψ are not well defined individually, so we do not have a one-to-one mapping (or a one-to-two mapping) between the 3-torus of (θ, φ, ψ) and S³.

If the Hopf fibration is treated as a vector field in 3 dimensional space then there is a solution to the (compressible, non-viscous) Navier–Stokes equations of fluid dynamics in which the fluid flows along the circles of the projection of the Hopf fibration in 3 dimensional space. The size of the velocities, the density and the pressure can be chosen at each point to satisfy the equations. All these quantities fall to zero going away from the centre. If a is the distance to the inner ring, the velocities, pressure and density fields are given by:

${\displaystyle \mathbf {v} (x,y,z)=A\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-2}\left(2(-ay+xz),2(ax+yz),a^{2}-x^{2}-y^{2}+z^{2}\right)}$

${\displaystyle p(x,y,z)=-A^{2}B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-3},}$

${\displaystyle \rho (x,y,z)=3B\left(a^{2}+x^{2}+y^{2}+z^{2}\right)^{-1}}$

for arbitrary constants A and B. Similar patterns of fields are found as soliton solutions of magnetohydrodynamics:[4]

A real version of the Hopf fibration is obtained by regarding the circle S¹ as a subset of R² in the usual way and by identifying antipodal points. This gives a fiber bundle S¹ → RP¹ over the real projective line with fiber S⁰ = {1, −1}. Just as CP¹ is diffeomorphic to a sphere, RP¹ is diffeomorphic to a circle.

More generally, the n-sphere Sⁿ fibers over real projective space RPⁿ with fiber S⁰.

The Hopf construction gives circle bundles p : S²ⁿ⁺¹ → CPⁿ over complex projective space. This is actually the restriction of the tautological line bundle over CPⁿ to the unit sphere in Cⁿ⁺¹.

Similarly, one can regard S⁴ⁿ⁺³ as lying in Hⁿ⁺¹ (quaternionic n-space) and factor out by unit quaternion (= S³) multiplication to get the quaternionic projective space HPⁿ. In particular, since S⁴ = HP¹, there is a bundle S⁷ → S⁴ with fiber S³.

A similar construction with the octonions yields a bundle S¹⁵ → S⁸ with fiber S⁷. But the sphere S³¹ does not fiber over S¹⁶ with fiber S¹⁵. One can regard S⁸ as the octonionic projective line OP¹. Although one can also define an octonionic projective plane OP², the sphere S²³ does not fiber over OP² with fiber S⁷.[5][6]

Sometimes the term "Hopf fibration" is restricted to the fibrations between spheres obtained above, which are

• S¹ → S¹ with fiber S⁰
• S³ → S² with fiber S¹
• S⁷ → S⁴ with fiber S³
• S¹⁵ → S⁸ with fiber S⁷

As a consequence of Adams's theorem, fiber bundles with spheres as total space, base space, and fiber can occur only in these dimensions. Fiber bundles with similar properties, but different from the Hopf fibrations, were used by John Milnor to construct exotic spheres.