Homotopy Groups of Spheres

Algebraic Topology · Axiom Academy

LESSON Homotopy Groups of Spheres One of the deepest and most challenging areas of algebraic topology The most fundamental result is that πₙ(Sⁿ) ≅ ℤ for all n ≥ 1. This can be proven using degree theory : every map f: Sⁿ → Sⁿ has an integer degree deg(f) that measures how many times the domain sphere wraps around the target sphere. The degree is a homotopy invariant: homotopic maps have the same degree The identity map has degree 1, constant maps have degree 0 The generator of πₙ(Sⁿ) is the homotopy class of the identity map. Maps of degree k correspond to k times the generator. 2. Below the Diagonal: πₖ(Sⁿ) = 0 for k < n When k < n, all maps from S k to S n are null-homotopic. This can be proven using several approaches: Give S n a CW structure with one 0-cell and one n-cell Any map f: S k → S n can be homotoped to a cellular map Since k < n, the image must lie in the 0-skeleton (a point) This result is intuitive: a lower-dimensional sphere has "room to contract" inside a higher-dimensional sphere without wrapping around it. 3. Above the Diagonal: π₃(S²) ≅ ℤ (Hopf) The first major surprise: π₃(S²) ≅ ℤ, discovered by Heinz Hopf in 1931. This shows that a higher-dimensional sphere can "wrap around" a lower-dimensional one in a non-trivial way! The generator of π₃(S²) comes from the Hopf fibration h: S³ → S². This is a fiber bundle with fiber S¹: Geometrically, S³ lives in ℂ² and S² can be viewed as ℂP¹. The Hopf map sends (z₁, z₂) to [z₁ : z₂].

This is the written version of the interactive lesson above. See the full Algebraic Topology course.