![Hopf Fibration](https://upload.wikimedia.org/wikipedia/commons/b/b9/Hopf_Fibration.png)
In the mathematical field of
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a
3-sphere (a
hypersphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
in
four-dimensional space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
) in terms of
circles
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
and an ordinary
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. Discovered by
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
Early life and education
Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...
in 1931, it is an influential early example of a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. Technically, Hopf found a many-to-one
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
(or "map") from the -sphere onto the -sphere such that each distinct ''point'' of the -sphere is mapped from a distinct
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
of the -sphere . Thus the -sphere is composed of fibers, where each fiber is a circle — one for each point of the -sphere.
This fiber bundle structure is denoted
:
meaning that the fiber space (a circle) is
embedded in the total space (the -sphere), and (Hopf's map) projects onto the base space (the ordinary -sphere). The Hopf fibration, like any fiber bundle, has the important property that it is
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
a
product space
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. However it is not a ''trivial'' fiber bundle, i.e., is not ''globally'' a product of and 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
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 o ...
are not trivial in general. It also provides a basic example of a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
, by identifying the fiber with the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
.
Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of the Hopf fibration induces a remarkable structure on , in which all of 3-dimensional space, except for the z-axis, is filled with nested
tori made of linking
Villarceau circles
In geometry, Villarceau circles () are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the e ...
. Here each fiber projects to a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
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
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of a circle of latitude of the -sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When 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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to circles, although they are not geometric
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
s.
There are numerous generalizations of the Hopf fibration. The unit sphere in
complex coordinate space
In mathematics, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-tuples of complex numbers. It is denoted \Complex^n, and is the ''n''-fold Cartesian product of the complex plane \Complex wi ...
fibers naturally over the
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 ...
with circles as fibers, and there are also
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
,
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
ic,
[quaternionic Hopf Fibration, ncatlab.org. https://ncatlab.org/nlab/show/quaternionic+Hopf+fibration] and
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
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:
:
:
:
:
By
Adams's theorem such fibrations can occur only in these dimensions.
The Hopf fibration is important in
twistor theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena ...
.
Definition and construction
For any
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''n'', an ''n''-dimensional sphere, or
n-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, cal ...
, can be defined as the set of points in an
-dimensional
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
which are a fixed distance from a central
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
. For concreteness, the central point can be taken to be the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
, 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,
, consists of the points
in
with ''x''
12 + ''x''
22 + ⋯+ ''x''
''n'' + 12 = 1. For example, the -sphere consists of the points (''x''
1, ''x''
2, ''x''
3, ''x''
4) in R
4 with ''x''
12 + ''x''
22 + ''x''
32 + ''x''
42 = 1.
The Hopf fibration of the -sphere over the -sphere can be defined in several ways.
Direct construction
Identify with and with (where denotes the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s) by writing:
:
and
:
.
Thus is identified with the
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of all in such that , and is identified with the subset of all in such that . (Here, for a complex number , where the star denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.) Then the Hopf fibration is defined by
:
The first component is a complex number, whereas the second component is real. Any point on the -sphere must have the property that . If that is so, then lies on the unit -sphere in , as may be shown by squaring the complex and real components of
:
Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if , then must equal for some complex number with . The converse is also true; any two points on the -sphere that differ by a common complex factor map to the same point on the -sphere. These conclusions follow, because the complex factor cancels with its complex conjugate in both parts of : in the complex component and in the real component .
Since the set of complex numbers with form the unit circle in the complex plane, it follows that for each point in , the
inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
is a circle, i.e., . Thus the -sphere is realized as a
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (th ...
of these circular fibers.
A direct parametrization of the -sphere employing the Hopf map is as follows.
:
:
or in Euclidean
:
:
:
:
Where runs over the range to , runs over the range and and can take any values between and . Every value of , except and which specify circles, specifies a separate
flat torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
in the -sphere, and one round trip ( to ) of either or causes you to make one full circle of both limbs of the torus.
A mapping of the above parametrization to the -sphere is as follows, with points on the circles parametrized by .
:
:
:
Geometric interpretation using the complex projective line
A geometric interpretation of the fibration may be obtained using the
complex projective line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers p ...
, , which is defined to be the set of all complex one-dimensional
subspaces of . Equivalently, is the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of by the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
which identifies with for any nonzero complex number ''λ''. On any complex line in C
2 there is a circle of unit norm, and so the restriction of the
quotient map
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
to the points of unit norm is a fibration of over .
is diffeomorphic to a -sphere: indeed it can be identified with the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
, which is the
one point compactification of (obtained by adding a
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adj ...
). The formula given for above defines an explicit diffeomorphism between the complex projective line and the ordinary -sphere in -dimensional space. Alternatively, the point can be mapped to the ratio in the Riemann sphere .
Fiber bundle structure
The Hopf fibration defines a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
, with bundle projection . This means that it has a "local product structure", in the sense that every point of the -sphere has some
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
whose inverse image in the -sphere can be
identified with the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of and a circle: . Such a fibration is said to be
locally trivial
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
.
For the Hopf fibration, it is enough to remove a single point from and the corresponding circle from ; thus one can take , and any point in 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 -sphere in ordinary -dimensional space. The
rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
has a
double cover, the
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a L ...
,
diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...
to the -sphere. The spin group acts
transitively on by rotations. The
stabilizer of a point is isomorphic to the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
. It follows easily that the -sphere is a
principal circle bundle over the -sphere, and this is the Hopf fibration.
To make this more explicit, there are two approaches: the group can either be identified with the group
Sp(1)
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
of
unit quaternion
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s, or with the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
.
In the first approach, a vector in is interpreted as a quaternion by writing
:
The -sphere is then identified with the
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s, the quaternions of unit norm, those for which , where , which is equal to for as above.
On the other hand, a vector in can be interpreted as an imaginary quaternion
:
Then, as is well-known since , the mapping
:
is a rotation in : indeed it is clearly an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, since , and it is not hard to check that it preserves orientation.
In fact, this identifies the group of
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s with the group of rotations of , modulo the fact that the versors and determine the same rotation. As noted above, the rotations act transitively on , and the set of versors which fix a given right versor have the form , where and are real numbers with . This is a circle subgroup. For concreteness, one can take , and then the Hopf fibration can be defined as the map sending a versor . All the quaternions , where is one of the circle of versors that fix , get mapped to the same thing (which happens to be one of the two rotations rotating to the same place as does).
Another way to look at this fibration is that every versor ω moves the plane spanned by to a new plane spanned by . Any quaternion , where is one of the circle of versors that fix , will have the same effect. We put all these into one fibre, and the fibres can be mapped one-to-one to the -sphere of rotations which is the range of .
This approach is related to the direct construction by identifying a quaternion with the matrix:
:
This identifies the group of versors with , and the imaginary quaternions with the skew-hermitian matrices (isomorphic to ).
Explicit formulae
The rotation induced by a unit quaternion is given explicitly by the
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
:
Here we find an explicit real formula for the bundle projection by noting that the fixed unit vector along the axis, , rotates to another unit vector,
:
which is a continuous function of . That is, the image of is the point on the -sphere where it sends the unit vector along the axis. The fiber for a given point on 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 in . Multiplication of unit quaternions produces composition of rotations, and
:
is a rotation by around the axis. As varies, this sweeps out a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
of , our prototypical fiber. So long as the base point, , is not the antipode, , the quaternion
:
will send to . Thus the fiber of is given by quaternions of the form , which are the points
:
Since multiplication by acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle.
The final fiber, for , can be given by defining to equal , producing
:
which completes the bundle. But note that this one-to-one mapping between and is not continuous on this circle, reflecting the fact that is not topologically equivalent to .
Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the -sphere is equivalent to a
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
, 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 vector) to all possible points on a unit -sphere. However, fixing the tip of the vector does not specify the rotation fully; a further rotation is possible about the axis. Thus, the -sphere is mapped onto the -sphere, plus a single rotation.
The rotation can be represented using the
Euler angles
The Euler angles are three angles introduced by Leonhard Euler to describe the Orientation (geometry), orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189 ...
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
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two ...
of (θ, φ, ψ) and ''S''
3.
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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
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 and . Similar patterns of fields are found as
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
solutions of
magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...
:
Generalizations
The Hopf construction, viewed as a fiber bundle ''p'': ''S''
3 → CP
''1'', admits several generalizations, which are also often known as Hopf fibrations. First, one can replace the projective line by an ''n''-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. Second, one can replace the complex numbers by any (real)
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
, including (for ''n'' = 1) the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s.
Real Hopf fibrations
A real version of the Hopf fibration is obtained by regarding the circle ''S''
1 as a subset of R
2 in the usual way and by
identifying antipodal points. This gives a fiber bundle ''S''
1 → RP
1 over the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line (geometry), line that has been historically introduced to solve a problem set by visual perspective (visual), perspect ...
with fiber ''S''
0 = . Just as CP
1 is diffeomorphic to a sphere, RP
1 is diffeomorphic to a circle.
More generally, the ''n''-sphere ''S''
''n'' fibers over
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
RP
''n'' with fiber ''S''
0.
Complex Hopf fibrations
The Hopf construction gives circle bundles ''p'' : ''S''
2''n''+1 → CP
''n'' over
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 ...
. This is actually the restriction of the
tautological line bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...
over CP
''n'' to the unit sphere in C
''n''+1.
Quaternionic Hopf fibrations
Similarly, one can regard ''S''
4''n+3'' as lying in H
''n+1'' (
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
ic ''n''-space) and factor out by unit quaternion (= ''S''
3) multiplication to get the
quaternionic projective space HP
''n''. In particular, since ''S''
4 = HP
1, there is a bundle ''S''
7 → ''S''
4 with fiber ''S''
3.
Octonionic Hopf fibrations
A similar construction with the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s yields a bundle ''S''
15 → ''S''
8 with fiber ''S''
7. But the sphere ''S''
31 does not fiber over ''S''
16 with fiber ''S''
15. One can regard ''S''
8 as the
octonionic projective line OP
1. Although one can also define an
octonionic projective plane
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.Baez (2002).
The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing ...
OP
2, the sphere ''S''
23 does not fiber over OP
2
with fiber ''S''
7.
Fibrations between spheres
Sometimes the term "Hopf fibration" is restricted to the fibrations between spheres obtained above, which are
* ''S''
1 → ''S''
1 with fiber ''S''
0
* ''S''
3 → ''S''
2 with fiber ''S''
1
* ''S''
7 → ''S''
4 with fiber ''S''
3
* ''S''
15 → ''S''
8 with fiber ''S''
7
As a consequence of
Adams's theorem, fiber bundles with
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s 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
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
to construct
exotic spheres.
Geometry and applications
The Hopf fibration has many implications, some purely attractive, others deeper. For example,
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
''S''
3 → R
3 induces a remarkable structure in R
3, which in turn illuminates the topology of the bundle . Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in R
3 which fill space. Here there is one exception: the Hopf circle containing the projection point maps to a straight line in R
3 — a "circle through infinity".
The fibers over a circle of latitude on ''S''
2 form a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
in ''S''
3 (topologically, a torus is the product of two circles) and these project to nested
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
es in R
3 which also fill space. The individual fibers map to linking
Villarceau circles
In geometry, Villarceau circles () are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the e ...
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 R
3 and in ''S''
3. Two such linking circles form a
Hopf link
In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf.
Geometric realization
A concrete model consists o ...
in R
3
Hopf proved that the Hopf map has
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
__TOC__ Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map''
:\eta\colon S^3 \to S ...
1, and therefore is not
null-homotopic
In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed ...
. In fact it generates the
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
π
3(''S''
2) and has infinite order.
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Riemann sphere is known as the
Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
, and the Hopf fibration describes the topological structure of a quantum mechanical
two-level system
In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a syst ...
or
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
. Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration
:
. Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the
Dirac monopole
In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
.
Hopf fibration also found applications in
robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
, where it was used to generate uniform samples on
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
for the
probabilistic roadmap algorithm in motion planning. It also found application in the
automatic control
Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines ...
of
quadrotors.
Notes
References
* ; reprinted as article 20 in
*
*
*
*.
*
*
*
External links
* {{springer, title=Hopf fibration, id=p/h047980
Dimensions MathChapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
An Elementary Introduction to the Hopf Fibrationby David W. Lyons (
PDF
Portable Document Format (PDF), standardized as ISO 32000, is a file format developed by Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, hardware, and operating systems. ...
)
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-cellfrom http://page.math.tu-berlin.de/~gunn/.
Interactive visualization of the mapping of points on the 2-sphere to circles in the 3-sphere
Algebraic topology
Geometric topology
Differential geometry
Fiber bundles
Homotopy theory