Hopf bundle - AbsoluteAstronomy.com

In the mathematical field of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere

3-sphere

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...

(a hypersphere

Hypersphere

In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...

in four-dimensional space) in terms of circles and an ordinary sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. Discovered by Heinz Hopf

Heinz Hopf

Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...

in 1931, it is an influential early example of a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

. Technically, Hopf found a many-to-one continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

(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 . 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

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

Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting...

are not trivial in general. It also provides a basic example of a principal bundle

Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...

, by identifying the fiber with the circle group.

Stereographic projection

Stereographic projection

The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...

of the Hopf fibration induces a remarkable structure on R³, in which space is filled with nested tori

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

made of linking Villarceau circles

Villarceau circles

In geometry, Villarceau circles are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in the plane parallel to the equatorial plane of the torus. Another is...

. 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 tori 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

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

, quaternion

Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...

ic, and octonion

Octonion

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...

ic 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:

In fact these are the only such fibrations between spheres, which fact is related to Hurwitz's Theorem.

The Hopf fibration is important in twistor theory

Twistor theory

In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3+1 space-time into geometric objects in a 4 dimensional space with metric signature...

Definition and construction

For any natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an (n+1)-dimensional space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

, 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.

Direct construction

Identify R⁴ with C² and R³ with C×R (where C denotes the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s) by writing: as (z₀ = x₁ + ix₂, z₁ = x₃ + ix₄); and as (z = x₁ + ix₂, x = x₃).
Thus S³ is identified with the subset

Subset

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

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

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

.) 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

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

Disjoint union

In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...

of these circular fibers.

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

Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

of C²\{0} by the equivalence relation

Equivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

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

Riemann sphere

In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

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_∞.

Fiber bundle structure

The Hopf fibration defines a fiber bundle

Fiber bundle

In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...

, 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

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

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.

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

Rotation group

In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...

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 quaternion

Quaternion

s, or with the special unitary group

Special unitary group

The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

SU(2).

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

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

Then, as is well-known since , the mapping

is a rotation in R³: indeed it is clearly an isometry

Isometry

In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

, 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 q which fix a given unit imaginary quaternion 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 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:

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).

Explicit formulae

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

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,

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

is a rotation by 2θ around the z axis. As θ varies, this sweeps out a great circle

Great circle

A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

of S³, our prototypical fiber. So long as the base point, (a,b,c), is not the antipode, (0,0,−1), the quaternion

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

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

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

Quaternion

, which in turn is equivalent to a particular rotation of a Cartesian coordinate frame

Cartesian coordinate system

A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

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.

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

Navier-Stokes equations

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...

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:

for arbitrary constants A and B. Similar patterns of fields are found as soliton

Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

solutions of magnetohydrodynamics

Magnetohydrodynamics

Magnetohydrodynamics is an academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water or electrolytes...

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

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

. Second, one can replace the complex numbers by any (real) division algebra

Division algebra

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...

, including (for n = 1) the octonion

Octonion

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

Real projective space

In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...

RPⁿ with fiber S⁰.

Complex Hopf fibrations

The Hopf construction gives circle bundles p : S²ⁿ⁺¹ → CPⁿ over complex projective space

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

. 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⁻¹ as lying in Hⁿ (quaternion

Quaternion

ic 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.

Octonionic Hopf fibrations

A similar construction with the octonion

Octonion

s 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

Cayley plane

In mathematics, the Cayley plane OP2 is a projective plane over the octonions. It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley ....

, OP², S³¹ does not fiber over it.

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 sphere

Sphere

s as total space, base space, and fiber.

Geometry and applications

The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection

Stereographic projection

of S³ to R³ induces a remarkable structure in R³, which in turn illuminates the topology of the bundle . 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

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

in S³ (topologically, a torus is the product of two circles) and these project to nested torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

es in R³ which also fill space. The individual fibers map to linking Villarceau circles

Villarceau circles

on these tori, with the exception of the circle through the projection point and the one through its opposite point

Antipodal point

In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....

: 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

Hopf link

thumb|right|[[Skein relation]] for the Hopf link.In mathematical knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once...

in R³

Hopf proved that the Hopf map has Hopf invariant

Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...

1, and therefore is not null-homotopic. In fact it generates the homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

π₃(S²) and has infinite order.

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the Riemann sphere is known as the Bloch sphere

Bloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

, and the Hopf fibration describes the topological structure of a quantum mechanical two-level system

Two-level system

In quantum mechanics, a two-state system is a system which has two possible states. More formally, the Hilbert space of a two-state system has two degrees of freedom, so a complete basis spanning the space must consist of two independent states...

or qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration

.
.

Discrete examples

The regular 4-polytopes

Convex regular 4-polytope

In mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....

: 8-cell (Tesseract

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

), 24-cell, and 120-cell, can each be partitioned into disjoint great circle rings of cells forming discrete Hopf fibrations of these polytopes. The Tesseract partitions into two interlocking rings of four cubes each. The 24-cell partitions into four rings of six octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

s each. The 120-cell partitions into twelve rings of ten dodecahedrons each.

External links

Dimensions Math Chapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
YouTube animation of the construction of the 120-cell By Gian Marco Todesco shows the Hopf fibration of the 120-cell.

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