πn is Abelian for n ≥ 2

Algebraic Topology · Axiom Academy

LESSON Higher Homotopy Groups are Abelian The Eckmann-Hilton argument proves that π n is abelian for n ≥ 2 The fundamental group π₁(X, x₀) consists of loops based at x₀. When we compose two loops f and g, the order matters: f * g traces f first, then g. Problem: In one dimension, loops are "stuck" in their order. We cannot slide f past g without leaving the space. This is why π₁ can be non-abelian. 2. Higher Homotopy Groups: Room to Slide For n ≥ 2, elements of π n (X) are represented by maps f: S n → X (or equivalently, f: I n → X with boundary mapped to basepoint). Key insight: When n ≥ 2, we have at least two dimensions. This gives us enough room to continuously deform one map "around" another. 3. The Eckmann-Hilton Argument The Eckmann-Hilton argument is an elegant proof technique that works whenever you have two binary operations ∗ and ⊕ that are compatible. Setup: For π n with n ≥ 2, we can compose maps in two different "directions": Horizontal composition (∗): Compose in the first coordinate Vertical composition (⊕): Compose in the second coordinate Both have identity elements, and they "interchange": (a ∗ b) ⊕ (c ∗ d) = (a ⊕ c) ∗ (b ⊕ d) 4. The Operations Agree and Are Abelian Step 1: Show ∗ = ⊕ using the interchange law with identities: Let e be the identity. Then: a ∗ b = (a ∗ e) ⊕ (e ∗ b) = (a ⊕ e) ∗ (e ⊕ b) = a ⊕ b Step 2: Show the operation is commutative: a ∗ b = (a ∗ b) ⊕ (e ∗ e) = (a ⊕ e) ∗ (b ⊕ e) = a ∗ b

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