Hopf fibration

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The Hopf fibration can be visualized using a stereographic projection of S³ into R³. Here keyrings mimick part of the Hopf fibration by showing some of the circles of the Hopf fibration which lie on a common torus.

In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere comes from a distinct circle of the 3-sphere (Hopf 1931). Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.

This fiber bundle structure is denoted

$S^1 \hookrightarrow S^3 \xrightarrow{\ p \, } S^2,$

where S³ (the 3-sphere) is the total space, S² (the ordinary 2-sphere) the base space, S¹ (a circle) the fiber space, and p: S³→S² (Hopf's map) the bundle projection. The Hopf fibration, like any fiber bundle, has the important property that it is locally a product space. However it is not a trivial fiber bundle, i.e., S³ is not (globally) a product of S² and S¹. This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group.

Stereographic projection of the Hopf fibration induces a remarkable structure on R³, in which space is filled with nested toruses made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a "circle through infinity" — a line). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) One of these toruses is illustrated by the image of linking keyrings on the right.

There are numerous generalizations of the Hopf fibration. The unit sphere in Cⁿ⁺¹ fibers naturally over CPⁿ with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres:

$S^0\hookrightarrow S^1 \rightarrow S^1, \,\!$

$S^1\hookrightarrow S^3 \rightarrow S^2, \,\!$

$S^3\hookrightarrow S^7\rightarrow S^4 \,\!$

$S^7\hookrightarrow S^{15}\rightarrow S^8. \,\!$

In fact these are the only such fibrations between spheres.

1 Definition and construction
- 1.1 Direct construction
  - 1.1.1 Geometric interpretation using the complex projective line
  - 1.1.2 Fiber bundle structure
- 1.2 Geometric interpretation using rotations
  - 1.2.1 Explicit formulae
2 Generalizations
3 Geometry and applications
4 References

[edit] Definition and construction

For any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an (n+1)-dimensional space which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the n-sphere, Sⁿ, consists of the points (x₁, x₂, …, x_n+1) in Rⁿ⁺¹ with x₁² + x₂² + ⋯+ x_n+1² = 1. For example, the 3-sphere consists of the points (x₁, x₂, x₃, x₄) in R⁴ with x₁² + x₂² + x₃² + x₄² = 1.

The Hopf fibration p: S³ → S² of the 3-sphere over the 2-sphere can be defined in several ways.

[edit] Direct construction

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

(x₁, x₂, x₃, x₄) as (z₀ = x₁ + ix₂, z₁ = x₃ + ix₄); and

(x₁, x₂, x₃) as (z = x₁ + ix₂, x = x₃).

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

p(z₀, z₁) = (2z₀z₁^∗, |z₀|² − |z₁|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 squaring the complex and real components of p

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

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.

[edit] Geometric interpretation using the complex projective line

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_∞.

[edit] Fiber bundle structure

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.

[edit] Geometric interpretation using rotations

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. 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

$q = x_1+\bold{i}x_2+\bold{j}x_3+\bold{k}x_4.\,\!$

The 3-sphere is then identified with the quaternions of unit norm, i.e., 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 an imaginary quaternion

$p = \bold{i}y_1+\bold{j}y_2+\bold{k}y_3. \,\!$

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

$p \mapsto q p q^* \,\!$

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 unit quaternions with the group of rotations of R³, modulo the fact that the unit quaternions q and −q determine the same rotation. As noted above, the rotations act transitively on S², and the set of unit quaternions p which fix a given unit imaginary quaternion p have the form 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 unit quaternion q to q k q^∗.

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:

$\begin{bmatrix} x_1+\bold i x_2 & x_3+\bold i x_4 \\ -x_3+\bold i x_4 & x_1-\bold i x_2 \end{bmatrix}.\,\!$

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

[edit] Explicit formulae

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

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

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

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

which is a continuous function of (w,x,y,z). That is, the image of q is where it aims the z axis. The fiber for a given point on S² consists of all those unit quaternions that aim there.

To write an explicit formula for the fiber over a point (a,b,c) in S², we may proceed as follows. Multiplication of unit quaternions produces composition of rotations, and

$q_{\theta} = \cos \theta + \bold{k} \sin \theta \,\!$

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

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

will aim there. 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

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

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 using q_(0,0,−1) = i, producing

$\Big(0,\cos (\theta ),-\sin (\theta ),0\Big) , \,\!$

which completes the bundle.

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.

[edit] Generalizations

The Hopf construction, viewed as a fiber bundle p: S³ → CP¹, admits several generalizations, which are also often known as Hopf fibrations. First, one can replace the projective line by an n-dimensional projective space. Second, one can replace the complex numbers by any (real) division algebra, including (for n = 1) the octonions.

[edit] Real Hopf fibrations

A real version of the Hopf fibration is obtained by regarding S¹ as a subset of R² in the usual way and factoring out by unit real multiplication to obtain and 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.

[edit] Complex Hopf fibrations

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.

[edit] Quaternionic Hopf fibrations

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

[edit] Octonionic Hopf fibrations

A similar construction with the octonions yields a bundle S¹⁵ → S⁸ with fiber S⁷. One can regard S⁸ as the octonionic projective line OP¹.

Although one can also define an octonionic projective plane, OP², S³¹ does not fiber over it.

[edit] Fibrations between spheres

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' theorem, these are the only fiber bundles with spheres as total space, base space, and fiber.

[edit] Geometry and applications

The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection of S³ to R³ induces a remarkable structure in R³, which in turn illuminates the topology of the bundle (Lyons 2003). Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in R³ which fill space. Here there is one exception: the Hopf circle containing the projection point maps to a straight line in R³ — a "circle through infinity".

The fibers over a circle of latitude on S² form a torus in S³ (topologically, a torus is the product of two circles) and these project to nested toruses in R³ which also fill space. The individual fibers map to linking Villarceau circles on these tori, with the exception of the circle through the projection point and the one through its opposite point: the former maps to a straight line, the latter to a unit circle perpendicular to, and centered on, this line, which may be viewed as a degenerate torus whose radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through every circle, both in R³ and in S³. Two such linking circles form a Hopf link in R³

Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic. In fact it generates the homotopy group π₃(S²) and has infinite order.

In quantum mechanics, the Riemann sphere is known as the Bloch sphere, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system or qubit. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration

$S^3 \hookrightarrow S^7\to S^4 \,\!$ .