Functoriality of π₁

Algebraic Topology · Axiom Academy

How the fundamental group transforms continuous maps into group homomorphisms A continuous map φ: (X, x₀) → (Y, y₀) between pointed spaces induces a group homomorphism φ₊: π₁(X, x₀) → π₁(Y, y₀). The map φ "pushes forward" loops in X to loops in Y by composition. The induced map φ₊ is well-defined: if f ≃ g (homotopic loops in X), then φ ∘ f ≃ φ ∘ g (homotopic loops in Y). This ensures that φ₊ maps homotopy classes to homotopy classes, not just loops to loops. The induced map φ₊ preserves the group operation: φ₊([f] · [g]) = φ₊([f]) · φ₊([g]). This makes φ₊ a group homomorphism, preserving the algebraic structure. For continuous maps φ: (X, x₀) → (Y, y₀) and ψ: (Y, y₀) → (Z, z₀), we have (ψ ∘ φ)₊ = ψ₊ ∘ φ₊. Computing both sides: (ψ ∘ φ)₊([f]) = [ψ ∘ φ ∘ f] = ψ₊([φ ∘ f]) = ψ₊(φ₊([f])) = (ψ₊ ∘ φ₊)([f]). The identity map id: (X, x₀) → (X, x₀) induces the identity homomorphism: id₊ = id on π₁(X, x₀). For any loop [f], we have id₊([f]) = [id ∘ f] = [f], so id₊ is indeed the identity homomorphism. The fundamental group π₁ defines a functor from Top to Grp (pointed topological spaces to groups). Morphisms: φ: (X, x₀) → (Y, y₀) ↦ φ₊: π₁(X, x₀) → π₁(Y, y₀) Preserves composition: (ψ ∘ φ)₊ = ψ₊ ∘ φ₊

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