Hopf bundle

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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:

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.

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

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 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

.

Contents 1 Geometry 2 Hopf map 3 Generalizations 3.1 Real, quaternionic, and octonionic Hopf bundles 4 References

[edit] 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,

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

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.

[edit] 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).

[edit] 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.

[edit] 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.

[edit] 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.