Higher Homotopy Groups Definition

Algebraic Topology · Axiom Academy

LESSON Higher Homotopy Groups Definition Extending the fundamental group to higher dimensions through based homotopy classes of spheres 1. Definition via Based Homotopy Classes For n ≥ 1, the nth homotopy group of a pointed space (X, x₀) is defined as the set of based homotopy classes of maps from the n-sphere to X. Here, [Sⁿ, X]₀ denotes the set of basepoint-preserving homotopy classes of continuous maps f: Sⁿ → X with f(s₀) = x₀, where s₀ is the basepoint of Sⁿ. 2. Alternative Definition via the n-Cube An equivalent formulation uses the n-dimensional unit cube Iⁿ = [0,1]ⁿ instead of the n-sphere. We consider maps that send the entire boundary ∂Iⁿ to the basepoint x₀. where f: Iⁿ → X satisfies f(∂Iⁿ) = x₀ . This means every point on the boundary of the cube maps to the basepoint. These two definitions are equivalent because the n-sphere Sⁿ is homeomorphic to the quotient space Iⁿ/∂Iⁿ (the n-cube with its boundary collapsed to a point). For n ≥ 1, πₙ(X, x₀) has a natural group structure. The operation is defined by concatenation along the first coordinate of the n-cube. This splits the first coordinate: the left half [0, 1/2] applies f (stretched to fill), and the right half [1/2, 1] applies g (stretched to fill). Identity: The constant map c(t) = x₀ for all t ∈ Iⁿ. Inverse: [f]⁻¹ = [f̄] where f̄(t₁, t₂, ..., tₙ) = f(1 - t₁, t₂, ..., tₙ) (reverse the first coordinate). 4. Visualizing Higher-Dimensional Concatenation

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