The Fundamental Group

Algebraic Topology · Axiom Academy

Definition of π₁(X,x₀) and understanding loops based at a point A loop is a path that starts and ends at the same point. γ(0) = x₀ (starts at the basepoint) γ(1) = x₀ (ends at the basepoint) The basepoint x₀ is a fixed reference point in the space. We study all loops that start and end at this point, allowing us to compose loops by concatenation. The fundamental group consists of homotopy classes of loops based at x₀ . π₁(X, x₀) = [γ] : γ is a loop based at x₀ Two loops γ₀ and γ₁ represent the same element in π₁(X, x₀) if and only if they are path-homotopic (can be continuously deformed into each other while keeping x₀ fixed). The group operation in π₁(X, x₀) comes from concatenating loops . (γ₁ · γ₂)(t) = γ₁(2t) for 0 ≤ t ≤ 1/2 γ₂(2t−1) for 1/2 ≤ t ≤ 1 First traverse γ₁ at double speed (covering it in time [0, 1/2]), then traverse γ₂ at double speed (in time [1/2, 1]). The result is a loop based at x₀ . The fundamental group satisfies all the axioms of a group under concatenation. Closure: Concatenation [γ₁] · [γ₂] is well-defined and gives another homotopy class of loops Associativity: ([γ₁] · [γ₂]) · [γ₃] = [γ₁] · ([γ₂] · [γ₃]) (up to homotopy) Identity: The constant loop e(t) = x₀ gives identity element [e], satisfying [e] · [γ] = [γ] · [e] = [γ] Inverses: For each [γ], the reverse loop γ̄(t) = γ(1−t) gives [γ]⁻¹, satisfying [γ] · [γ̄] = [γ̄] · [γ] = [e]

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