24
45°
30
4
Hopf Fibration
Heinz Hopf 1931: S³→S² with fiber S¹. Every point on S² lifts to a great circle on S³. Two fibers over distinct base points are linked exactly once (Hopf invariant = 1). Stereographic projection h:S³→ℝ³ maps each circle-fiber to a torus knot. Fibers over a great circle on S² form a single torus (Clifford torus). Underpins quantum mechanics (spin-½ spinors live on S³) and is the simplest non-trivial principal bundle.
Heinz Hopf 1931: S³→S² with fiber S¹. Every point on S² lifts to a great circle on S³. Two fibers over distinct base points are linked exactly once (Hopf invariant = 1). Stereographic projection h:S³→ℝ³ maps each circle-fiber to a torus knot. Fibers over a great circle on S² form a single torus (Clifford torus). Underpins quantum mechanics (spin-½ spinors live on S³) and is the simplest non-trivial principal bundle.