Hopf fibration

h : S³ → S²

Stereographic projection into R³

Fibers of h : S³ → S² after σ from N = (0,0,0,1)

t = 0.000

r = √(1−t²) = 1.000

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Base space S²

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t 0.000

Slice height t t = 0.000

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The Hopf fibration

Spaces

S³ = { (z₁, z₂) ∈ ℂ² : |z₁|² + |z₂|² = 1 }

Writing z₁ = x + i y and z₂ = z + i t,

S³ = { (x, y, z, t) ∈ ℝ⁴ : x² + y² + z² + t² = 1 }

The base is the unit sphere S² ⊂ ℝ³.

Hopf map

h : S³ → S², h(z₁, z₂) = ( 2 Re(z₁ z̄₂), 2 Im(z₁ z̄₂), |z₁|² − |z₂|² )

Equivalently in real coordinates,

h(x, y, z, t) = ( 2(xz + yt), 2(yz − xt), x² + y² − z² − t² )

Fibers

The circle group S¹ acts on S³ by

e^iθ · (z₁, z₂) = (e^iθ z₁, e^iθ z₂)

This action preserves h, so each fiber h⁻¹(b) is an S¹-orbit (a circle in S³). With θ = 2πs and p₀ = (x₀, y₀, z₀, t₀),

(x, y, z, t)(s) = (x₀ cos θ − y₀ sin θ, x₀ sin θ + y₀ cos θ, z₀ cos θ − t₀ sin θ, z₀ sin θ + t₀ cos θ)

A point b on the lower-left S² selects the fiber h⁻¹(b); color matches the base point.

Stereographic projection

From the pole N = (0,0,0,1) (where t = 1),

σ(x, y, z, t) = ( x/(1−t), y/(1−t), z/(1−t) ) ∈ ℝ³

Fibers project to circles in ℝ³ (or a line, for the fiber through N). Distinct fibers remain linked.

Fixed-t slice

The slider chooses a height t ∈ [−1, 1]. The grey sphere is the level set

x² + y² + z² = 1 − t², r = √(1 − t²)

So at t = 0 one has the unit 2-sphere; at t = ±1 the slice collapses to a point (the poles (0,0,0,±1) of S³).

Markers are the intersections of each selected fiber with this level set. By construction every marker lies on the grey sphere. A generic fiber meets a given t in 0, 1, or 2 points (only when |t| ≤ |z₂| for that fiber).