The Hurewicz Theorem

Algebraic Topology · Axiom Academy

Connecting Homotopy and Homology: When Do They Agree? For any pointed space (X, x₀), there exists a natural homomorphism from the n-th homotopy group to the n-th homology group. It sends a homotopy class of maps [f: Sⁿ → X] to the homology class of the image of the fundamental class of Sⁿ under f. The Hurewicz map is constructed by taking a representative f: Sⁿ → X of an element in πₙ(X), viewing it as a singular n-cycle, and taking its homology class. Key Properties: The Hurewicz homomorphism is a group homomorphism that is natural with respect to continuous maps between pointed spaces. 2. Statement of the Hurewicz Theorem The Hurewicz theorem gives conditions under which the Hurewicz homomorphism is an isomorphism. Let (X, x₀) be a path-connected, pointed space that is (n-1)-connected (meaning πᵢ(X) = 0 for all i < n). The homology groups H̃ᵢ(X) = 0 for all i < n The Hurewicz homomorphism h: πₙ(X) → Hₙ(X) is an isomorphism For n = 1, if X is path-connected, the theorem states that the abelianization of π₁(X) is isomorphic to H₁(X). 3. First Non-Vanishing Groups Agreement A powerful consequence: the first non-trivial homotopy and homology groups appear at the same dimension and agree. Example: Consider the n-sphere Sⁿ (n ≥ 2): Sⁿ is (n-1)-connected: πᵢ(Sⁿ) = 0 for i < n By Hurewicz: πₙ(Sⁿ) ≅ Hₙ(Sⁿ) ≅ ℤ Both groups first appear at dimension n 4. Applications to Computing πₙ The Hurewicz theorem is a powerful computational tool for determining homotopy groups.

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