Introduction to the Fundamental Groups

1. 1.1. Loops and Fundamental group. Let X be an arcwise connected topological space. Consider the homotopy classes of loops starting at a fixed point b 0 ∈ X : Ω(X, b 0). Recall that a loop is a continuous mapping σ : (I, ∂I) → (X, b 0). Composition σ * τ of two loops σ, τ is defined as (σ * τ)(t) =    σ(2t), 0 ≤ t ≤ 1/2 τ (2t − 1), 1/2 ≤ t ≤ 1 Let c be the constant loop at b 0 and σ −1 be the loop defined by σ −1 (t) = σ(1 − t). Proposition 1. (1) (Well-defined up-to homotopy) σ ≃ σ ′ , τ ≃ τ ′ =⇒ σ * τ ≃ σ ′ * τ ′. (2) (Associativity) σ * (τ * ω) ≃ (σ * τ) * ω. (3) (Unit element) σ * c ≃ σ ≃ c * σ. (4) (Inverse) σ * σ −1 ≃ σ −1 * σ ≃ c. Example 3. π 1 (S 1 , b 0) ∼ = Z. π 1 (R 2 \ {o}) ∼ = Z. Proposition 4. b ′ 0 ∈ X, ℓ : [0, 1] → X, ℓ(0) = b 0 , ℓ(1) = b ′ 0. Theorem 5. The canonical homomorphism π 1 (X, b 0) → H 1 (X) is a surjective and the kernel is the commutator subgroup of π 1 (X, b 0).