Hopf fibration

File:Hopf Fibration.png

The Hopf fibration can be visualized using a stereographic projection of Template:Math to Template:Math and then compressing Template:Math to a ball. This image shows points on Template:Math and their corresponding fibers with the same color.

File:Hopfkeyrings.jpg

Pairwise linked keyrings mimic part of the Hopf fibration.

In differential 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 Template:Math-sphere onto the Template:Math-sphere such that each distinct point of the Template:Math-sphere is mapped from a distinct great circle of the Template:Math-sphere Template:Harv.<ref>This partition of the Template:Math-sphere into disjoint great circles is possible because, unlike with the Template:Math-sphere, distinct great circles of the Template:Math-sphere need not intersect.</ref> Thus the Template:Math-sphere is composed of fibers, where each fiber is a circle — one for each point of the Template:Math-sphere.

This fiber bundle structure is denoted

<math>S^1 \hookrightarrow S^3 \xrightarrow{\ p \, } S^2, </math>

meaning that the fiber space Template:Math (a circle) is embedded in the total space Template:Math (the Template:Math-sphere), and Template:Math (Hopf's map) projects Template:Math onto the base space Template:Math (the ordinary Template:Math-sphere). 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., Template:Math is not globally a product of Template:Math and Template:Math although locally it is indistinguishable from it.

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 Template:Math, in which all of 3-dimensional space, except for the z-axis, is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the Template:Math-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When Template:Math is compressed to the boundary of a ball, some geometric structure is lost although the topological structure is retained (see Topology and geometry). The loops are homeomorphic to circles, although they are not geometric circles.

There are numerous generalizations of the Hopf fibration. The unit sphere in complex coordinate space Template:Math fibers naturally over the complex projective space Template:Math with circles as fibers, and there are also real, quaternionic,<ref name="quaternionic Hopf Fibration on nLab">quaternionic Hopf Fibration, ncatlab.org. https://ncatlab.org/nlab/show/quaternionic+Hopf+fibration</ref> 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:

<math>S^0\hookrightarrow S^1 \to S^1,</math>

<math>S^1\hookrightarrow S^3 \to S^2,</math>

<math>S^3\hookrightarrow S^7 \to S^4,</math>

<math>S^7\hookrightarrow S^{15}\to S^8.</math>

By Adams's theorem such fibrations can occur only in these dimensions.

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 <math>(n+1)</math>-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, <math>S^n</math>, consists of the points <math>(x_1, x_2,\ldots , x_{n+ 1})</math> in <math>\R^{n+1}</math> with x₁² + x₂² + ⋯+ x_{n + 1}² = 1. For example, the Template:Math-sphere consists of the points (x₁, x₂, x₃, x₄) in R⁴ with x₁² + x₂² + x₃² + x₄² = 1.

The Hopf fibration Template:Math of the Template:Math-sphere over the Template:Math-sphere can be defined in several ways.

Direct construction

Identify Template:Math with Template:Math (where Template:Math denotes the complex numbers) by writing:

<math>(x_1, x_2, x_3, x_4) \leftrightarrow (z_0, z_1) = (x_1 + ix_2, x_3+ix_4),</math>

and identify Template:Math with Template:Math by writing

<math>(x_1, x_2, x_3) \leftrightarrow (z, x) = (x_1 + ix_2, x_3)</math>.

Thus Template:Math is identified with the subset of all Template:Math in Template:Math such that Template:Math, and Template:Math is identified with the subset of all Template:Math in Template:Math such that Template:Math. (Here, for a complex number Template:Math, its squared absolute value is |z|² = z z^∗ = x² + y², where the star denotes the complex conjugate.) Then the Hopf fibration Template:Math is defined by

<math>p(z_0,z_1) = (2z_0z_1^{\ast}, \left|z_0 \right|^2-\left|z_1 \right|^2).</math>

The first component is a complex number, whereas the second component is real. Any point on the Template:Math-sphere must have the property that Template:Math. If that is so, then Template:Math lies on the unit Template:Math-sphere in Template:Math, as may be shown by adding the squares of the absolute values of the complex and real components of Template:Math

<math>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_{1} \right|^{4} = \left( \left| z_{0} \right|^{2} + \left| z_{1} \right|^{2} \right)^{2} = 1</math>

Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if Template:Math, then Template:Math must equal Template:Math for some complex number Template:Math with Template:Math. The converse is also true; any two points on the Template:Math-sphere that differ by a common complex factor Template:Math map to the same point on the Template:Math-sphere. These conclusions follow, because the complex factor Template:Math cancels with its complex conjugate Template:Math in both parts of Template:Math: in the complex Template:Math component and in the real component Template:Math.

Since the set of complex numbers Template:Math with Template:Math form the unit circle in the complex plane, it follows that for each point Template:Math in Template:Math, the inverse image Template:Math is a circle, i.e., Template:Math. Thus the Template:Math-sphere is realized as a disjoint union of these circular fibers.

A direct parametrization of the Template:Math-sphere employing the Hopf map is as follows.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>z_0 = e^{i\,\frac{\xi_1+\xi_2}{2}}\sin\eta </math>

<math>z_1 = e^{i\,\frac{\xi_2-\xi_1}{2}}\cos\eta. </math>

or in Euclidean Template:Math

<math>x_1 = \cos\left(\frac{\xi_1+\xi_2}{2}\right)\sin\eta</math>

<math>x_2 = \sin\left(\frac{\xi_1+\xi_2}{2}\right)\sin\eta </math>

<math>x_3 = \cos\left(\frac{\xi_2-\xi_1}{2}\right)\cos\eta </math>

<math>x_4 = \sin\left(\frac{\xi_2-\xi_1}{2}\right)\cos\eta </math>

Where Template:Math runs over the range from Template:Math to Template:Math, Template:Math runs over the range from Template:Math to Template:Math, and Template:Math can take any value from Template:Math to Template:Math. Every value of Template:Math, except Template:Math and Template:Math which specify circles, specifies a separate flat torus in the Template:Math-sphere, and one round trip (Template:Math to Template:Math) of either Template:Math or Template:Math causes you to make one full circle of both limbs of the torus.

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

<math>z = \cos(2\eta)</math>

<math>x = \sin(2\eta)\cos\xi_1</math>

<math>y = \sin(2\eta)\sin\xi_1</math>

Geometric interpretation using the complex projective line

A geometric interpretation of the fibration may be obtained using the complex projective line, Template:Math, which is defined to be the set of all complex one-dimensional subspaces of Template:Math. Equivalently, Template:Math is the quotient of Template:Math by the equivalence relation which identifies Template:Math with Template:Math for any nonzero complex number Template:Math. On any complex line in Template:Math 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 Template:Math over Template:Math.

Template:Math is diffeomorphic to a Template:Math-sphere: indeed it can be identified with the Riemann sphere Template:Math, which is the one point compactification of Template:Math (obtained by adding a point at infinity). The formula given for Template:Math above defines an explicit diffeomorphism between the complex projective line and the ordinary Template:Math-sphere in Template:Math-dimensional space. Alternatively, the point Template:Math can be mapped to the ratio Template:Math in the Riemann sphere Template:Math.

Fiber bundle structure

The Hopf fibration defines a fiber bundle, with bundle projection Template:Math. This means that it has a "local product structure", in the sense that every point of the Template:Math-sphere has some neighborhood Template:Math whose inverse image in the Template:Math-sphere can be identified with the product of Template:Math and a circle: Template:Math. Such a fibration is said to be locally trivial.

For the Hopf fibration, it is enough to remove a single point Template:Math from Template:Math and the corresponding circle Template:Math from Template:Math; thus one can take Template:Math, and any point in Template:Math has a neighborhood of this form.

Geometric interpretation using rotations

Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the Template:Math-sphere in ordinary Template:Math-dimensional space. The rotation group SO(3) has a double cover, the spin group Template:Math, diffeomorphic to the Template:Math-sphere. The spin group acts transitively on Template:Math 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 Template:Math-sphere is a principal circle bundle over the Template:Math-sphere, and this is the Hopf fibration.

To make this more explicit, there are two approaches: the group Template:Math 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 Template:Math in Template:Math is interpreted as a quaternion Template:Math by writing

<math> q = x_1+\mathbf{i}x_2+\mathbf{j}x_3+\mathbf{k}x_4.\,\!</math>

The Template:Math-sphere is then identified with the versors, the quaternions of unit norm, those Template:Math for which Template:Math, where Template:Math, which is equal to Template:Math for Template:Math as above.

On the other hand, a vector Template:Math in Template:Math can be interpreted as a pure quaternion

<math> p = \mathbf{i}y_1+\mathbf{j}y_2+\mathbf{k}y_3. \,\!</math>

Then, as is well-known since Template:Harvtxt, the mapping

<math> p \mapsto q p q^* \,\!</math>

is a rotation in Template:Math: indeed it is clearly an isometry, since Template:Math, 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 Template:Math, modulo the fact that the versors Template:Math and Template:Math determine the same rotation. As noted above, the rotations act transitively on Template:Math, and the set of versors Template:Math which fix a given right versor Template:Math have the form Template:Math, where Template:Math and Template:Math are real numbers with Template:Math. This is a circle subgroup. For concreteness, one can take Template:Math, and then the Hopf fibration can be defined as the map sending a versor Template:Math. All the quaternions Template:Math, where Template:Math is one of the circle of versors that fix Template:Math, get mapped to the same thing (which happens to be one of the two Template:Math rotations rotating Template:Math to the same place as Template:Math does).

Another way to look at this fibration is that every versor ω moves the plane spanned by Template:Math to a new plane spanned by Template:Math. Any quaternion Template:Math, where Template:Math is one of the circle of versors that fix Template:Math, will have the same effect. We put all these into one fibre, and the fibres can be mapped one-to-one to the Template:Math-sphere of Template:Math rotations which is the range of Template:Math.

This approach is related to the direct construction by identifying a quaternion Template:Math with the Template:Math matrix:

<math>\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}.\,\!</math>

This identifies the group of versors with Template:Math, and the imaginary quaternions with the skew-hermitian Template:Math matrices (isomorphic to Template:Math).

Explicit formulae

The rotation induced by a unit quaternion Template:Math is given explicitly by the orthogonal matrix

<math>\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} . </math>

Here we find an explicit real formula for the bundle projection by noting that the fixed unit vector along the Template:Math axis, Template:Math, rotates to another unit vector,

<math> \Big(2(xz+wy) , 2(yz-wx) , 1-2(x^2+y^2)\Big) , \,\!</math>

which is a continuous function of Template:Math. That is, the image of Template:Math is the point on the Template:Math-sphere where it sends the unit vector along the Template:Math axis. The fiber for a given point on Template:Math 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 Template:Math in Template:Math. Multiplication of unit quaternions produces composition of rotations, and

<math>q_{\theta} = \cos \theta + \mathbf{k} \sin \theta</math>

is a rotation by Template:Math around the Template:Math axis. As Template:Math varies, this sweeps out a great circle of Template:Math, our prototypical fiber. So long as the base point, Template:Math, is not the antipode, Template:Math, the quaternion

<math> q_{(a,b,c)} = \frac{1}{\sqrt{2(1+c)}}(1+c-\mathbf{i}b+\mathbf{j}a) </math>

will send Template:Math to Template:Math. Thus the fiber of Template:Math is given by quaternions of the form Template:Math, which are the Template:Math points

<math> \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) . \,\!</math>

Since multiplication by Template:Math acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle.

The final fiber, for Template:Math, can be given by defining Template:Math to equal Template:Math, producing

<math> \Big(0,\cos (\theta ),-\sin (\theta ),0\Big),</math>

which completes the bundle. But note that this one-to-one mapping between Template:Math and Template:Math is not continuous on this circle, reflecting the fact that Template:Math is not topologically equivalent to Template:Math.

Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the Template:Math-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 Template:Math vector) to all possible points on a unit Template:Math-sphere. However, fixing the tip of the Template:Math vector does not specify the rotation fully; a further rotation is possible about the Template:Mathaxis. Thus, the Template:Math-sphere is mapped onto the Template:Math-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.

Fluid mechanics

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:

<math>\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)</math>

<math>p(x,y,z) = -A^2B \left(a^2+x^2+y^2+z^2\right)^{-3},</math>

<math>\rho(x,y,z) = 3B\left(a^2+x^2+y^2+z^2\right)^{-1}</math>

for arbitrary constants Template:Math and Template:Math. Similar patterns of fields are found as soliton solutions of magnetohydrodynamics:<ref>Template:Citation</ref>

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.

Real Hopf fibrations

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.

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.

Quaternionic Hopf fibrations

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.

Octonionic Hopf fibrations

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.<ref>Template:Cite book (§0.26 on page 6)</ref><ref>sci.math.research 1993 thread "Spheres fibred by spheres"</ref>

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'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.

Geometry and applications

File:Villarceau circles.gif

The fibers of the Hopf fibration stereographically project to a family of Villarceau circles in R³.

The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection S³ → R³ induces a remarkable structure in R³, which in turn illuminates the topology of the bundle Template:Harv. 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 minor 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

<math>S^3 \hookrightarrow S^7\to S^4.</math>

Template:Harv. Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the Dirac monopole.<ref>Template:Cite journal</ref>

Hopf fibration also found applications in robotics, where it was used to generate uniform samples on SO(3) for the probabilistic roadmap algorithm in motion planning.<ref>Template:Cite journal</ref> It also found application in the automatic control of quadrotors.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

See also

• Villarceau circles

Notes

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References

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External links

• Template:Springer
• Template:Mathworld
• Dimensions Math Chapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
• An Elementary Introduction to the Hopf Fibration by David W. Lyons (PDF)
• YouTube animation showing dynamic mapping of points on the 2-sphere to circles in the 3-sphere, by Professor Niles Johnson.
• YouTube animation of the construction of the 120-cell By Gian Marco Todesco shows the Hopf fibration of the 120-cell.
• Video of one 30-cell ring of the 600-cell from http://page.math.tu-berlin.de/~gunn/.
• Interactive visualization of the mapping of points on the 2-sphere to circles in the 3-sphere