In mathematics, the Hopf bundle (or Hopf fibration), named after Heinz Hopf, is an important example of a fiber bundle. It has base space S², total space S³, and fiber S¹: Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... A sphere is a perfectly symmetrical geometrical object. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... Illustration of a unit circle. ...

It was discovered by Heinz Hopf in 1931. The Hopf bundle also gives an example of a principal bundle by identifying the fiber with the circle group. Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany. ... Year 1931 (MCMXXXI) was a common year starting on Thursday (link is to a full 1931 calendar). ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1. ...

Key-Ring Model of the Hopf Fibration.

To construct the Hopf bundle, consider S³ as the subset of all (z₀, z₁) in C² such that |z₀|² + |z₁|² = 1. Identify (z₀, z₁) with (λz₀, λz₁) where λ is a complex number with norm one. Then the quotient of S³ by this equivalence relation is the complex projective line, CP¹, also known as the Riemann sphere S². Clearly the fiber of a point is S¹, and it is easy to show that local triviality holds, so that the Hopf bundle is a fiber bundle. The key-ring model in the picture can be mathematically described as a stereographic projection of S³ into R³. It does not show all the circles, of course (they would fill all of R³) but rather only those lying on a common torus in S³ Image File history File links Download high-resolution version (649x648, 321 KB) Originally uploaded to english wikipedia the 16:24, 28 November 2005 by Davidarichter, with this comment: This is a photograph of the key-ring model of the Hopf fibration. ... Image File history File links Download high-resolution version (649x648, 321 KB) Originally uploaded to english wikipedia the 16:24, 28 November 2005 by Davidarichter, with this comment: This is a photograph of the key-ring model of the Hopf fibration. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, an equivalence relation, denoted by an infix ~, is a binary relation on a set X that is reflexive, symmetric, and transitive. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... A rendering of the Riemann Sphere. ... Stereographic projection of a circle of radius R onto the x axis. ... In geometry, a torus (pl. ...

Another way to look at the Hopf bundle is to regard S³ as the special unitary group SU(2). The group SU(2) is isomorphic to Spin(3) and so acts transitively on S² by rotations. The stabilizer of a point is isomorphic to the circle group U(1). According to standard Lie group theory, SU(2) is then a principal U(1)-bundle over the left coset space SU(2)/U(1) which is diffeomorphic to the 2-sphere. The fibers in this bundle are just the left cosets of U(1) in SU(2). In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. ... In mathematics, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

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 Fig. ... Bloch sphere In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a 2-level quantum mechanical system. ... A two-level system is a quantum mechanical system in which there are two physical states of different energy. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...

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Geometry

We may also interpret the bundle projection S³→S² in terms of unit quaternions and 3D rotations. Let a point on S³ have coordinates (w,x,y,z). Interpret this as a quaternion, q, with unit quaternion norm, In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... A sphere rotating around its axis. ...

where N(q) = qq^* = w²+x²+y²+z². Interpret a vector (a,b,c) in R³ as the quaternion

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

is a rotation in R³, which we can express in matrix form as

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 a fiber, 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 of S³, our prototypical fiber. So long as the base point, (a,b,c), is not the antipode, (0,0,−1), the quaternion For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

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.

The bundle geometry induces a remarkable structure in R³, which in turn illuminates the topology of the bundle. Stereographic projection of S³ to R³ preserves circles and fills space. The fibers of a circle of latitude on S² form a torus in S³, and the individual fibers map to Villarceau circles in R³. There are two minor exceptions. One is the fiber containing the projection point, which maps to a line. The other is the fiber through the opposite point, where the torus shrinks to a unit circle perpendicular to, and centered on, the line. We can easily show (Lyons 2003) that 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. Stereographic projection of a circle of radius R onto the x axis. ... In geometry, a torus (pl. ... In geometry, given an arbitrary point on a torus, four circles can be drawn through it. ...

Hopf map

The Hopf map p : S³ → S² is defined by

p (z₀, z₁) = (|z₀|² - |z₁|², 2z₀z₁*)

the first component is a real number, the second complex, so together they define a point in R³. It's easy to check that if |z₀|² + |z₁|² = 1, then p (z₀, z₁) lies on the unit 2-sphere. Conversely, if p (z₀, z₁) = p (z₂, z₃) then (z₂, z₃) = (λz₀, λz₁) for some unit λ.

Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic, but is of infinite order in π₃(S²). In fact, the Hopf map generates π₃(S²). // Motivation The rational homotopy of an odd sphere ( odd) is zero unless . ... In mathematics, a continuous function from M to N is null-homotopic if it is homotopic to a constant function. ...

Generalizations

More generally, 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. In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ... The canonical or tautological line bundle on a projective space appears frequently in mathematics, often in the study of characteristic classes. ...

Real, quaternionic, and octonionic Hopf bundles

One may also regard S¹ as lying in R² and factor out by unit real multiplication to obtain RP¹ = S¹ and a fiber bundle S¹ → S¹ with fiber S⁰. Similarly, one can regard S⁴ⁿ⁻¹ as lying in Hⁿ (quaternionic n-space) and factor out by unit quaternion (= S³) multiplication to get HPⁿ. In particular, since S⁴ = HP¹, there is a bundle S⁷ → S⁴ with fiber S³. A similar construction with the octonions yields a bundle S¹⁵ → S⁸ with fiber S⁷. These bundles are sometimes also called Hopf bundles. As a consequence of Adams' theorem, these are the only fiber bundles with spheres as total space, base space, and fiber. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... A sphere is a perfectly symmetrical geometrical object. ...

References

• Cayley, Arthur (1845). "On certain results relating to quaternions". Philosophical Magazine 26: 141–145.
• Hopf, Heinz (1931). "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche" (PDF). Mathematische Annalen 104: 637–665. ISSN 0025-5831.
• Hopf, Heinz (1935). "Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension". Fundamenta Mathematicae 25: 427–440. ISSN 0016-2736.
• Lyons, David W. (April 2003). "An Elementary Introduction to the Hopf Fibration" (PDF). Mathematics Magazine 76 (2): 87–98. ISSN 0025-570X.