Hopf Fibration - Geometry and Applications

Geometry and Applications

The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection of S3 to R3 induces a remarkable structure in R3, which in turn illuminates the topology of the bundle (Lyons 2003). 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 in S3 (topologically, a torus is the product of two circles) and these project to nested toruses in R3 which also fill space. The individual fibers map to linking Villarceau circles 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 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 in R3

Hopf proved that the Hopf map has Hopf invariant 1, and therefore is not null-homotopic. In fact it generates the homotopy group π₃(S2) and has infinite order.

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

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(Mosseri & Dandoloff 2001).