The Hopf Fibration

Algebraic Topology · Axiom Academy

A remarkable fibration linking spheres of different dimensions A fibration is a continuous surjective map π: E → B where E is the total space and B is the base space . The fiber over a point b ∈ B is F = π⁻¹(b). Total space: S³ (the 3-sphere) Each fiber is a great circle in S³ Any two distinct fibers are linked once 2. Construction via Complex Numbers We can view S³ as unit vectors in ℂ², and S² as the complex projective line ℂℙ¹ (or the Riemann sphere). The fiber over a point [w] ∈ ℂℙ¹ consists of all (z₁, z₂) ∈ S³ such that z₁/z₂ = w (when z₂ ≠ 0). This fiber forms a circle: If w = z₁/z₂, then (z₁, z₂) = (wz₂, z₂) Constraint: |w|²|z₂|² + |z₂|² = 1, so |z₂| = 1/√(1 + |w|²) The fiber is e^(iθ)(w/√(1+|w|²), 1/√(1+|w|²)) : θ ∈ [0, 2π) Each point on S² corresponds to a circle (S¹) in S³. These circles have remarkable geometric properties: Disjoint: Fibers over different points never intersect Linked: Any two distinct fibers form a Hopf link (linking number = 1) Great circles: Each fiber is a geodesic in S³ Fills S³: The union of all fibers is exactly S³ Using stereographic projection from S³ to ℝ³, fibers appear as circles and lines (including the line at infinity). The north and south poles of S² correspond to fibers that project to particularly symmetric configurations. 4. Computing π₃(S²) from the Long Exact Sequence Every fibration F → E → B induces a long exact sequence (LES) of homotopy groups: For the Hopf fibration S¹ → S³ → S², we know:

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