Hopf fibration
Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle.
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 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 R3, in which all of 3-dimensional space, except for the z-axis, is filled with nested tori made of linking Villarceau circles.
Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere.
The unit sphere in complex coordinate space Cn+1 fibers naturally over the complex projective space CPn with circles as fibers, and there are also real, quaternionic,[2] and octonionic versions of these fibrations.
For any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an
The formula given for p above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space.
The Hopf fibration defines a fiber bundle, 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 with the product of U and a circle: p−1(U) ≅ U × S1.
Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the 2-sphere in ordinary 3-dimensional space.
On the other hand, a vector (y1, y2, y3) in R3 can be interpreted as a pure quaternion Then, as is well-known since Cayley (1845), the mapping is a rotation in R3: indeed it is clearly an isometry, 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.
But note that this one-to-one mapping between S3 and S2×S1 is not continuous on this circle, reflecting the fact that S3 is not topologically equivalent to S2×S1.
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.
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.
If a is the distance to the inner ring, the velocities, pressure and density fields are given by: for arbitrary constants A and B.
Second, one can replace the complex numbers by any (real) division algebra, including (for n = 1) the octonions.
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.
The Hopf construction gives circle bundles p : S2n+1 → CPn over complex projective space.
[5][6] Sometimes the term "Hopf fibration" is restricted to the fibrations between spheres obtained above, which are 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.
For example, stereographic projection S3 → R3 induces a remarkable structure in R3, which in turn illuminates the topology of the bundle (Lyons 2003).
Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration (Mosseri & Dandoloff 2001).
Moreover, the Hopf fibration is equivalent to the fiber bundle structure of the Dirac monopole.
[7] 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.
