Group Operation on Loops

Algebraic Topology · Axiom Academy

LESSON Group Operation on Loops Understanding how loop concatenation gives the fundamental group its algebraic structure 1. Loop Concatenation (Product Operation) Given two loops f and g based at x₀, we define their product f · g as the loop that traverses f first (at double speed) and then g (at double speed). The key insight: we reparametrize each loop to take half the time, then concatenate them. 2. Associativity (Up to Homotopy) The product is associative up to homotopy: (f · g) · h ≃ f · (g · h). While not equal as functions, they traverse the same path and are homotopic. This homotopy can be constructed explicitly by continuously reparametrizing the speeds at which we traverse each segment. The constant loop e(t) = x₀ for all t ∈ [0,1] serves as the identity element. For any loop f, we have e · f ≃ f ≃ f · e. The homotopy "erases" the constant portion by continuously speeding up the non-trivial part. Every loop f has an inverse f⁻¹, defined as the reverse loop: f⁻¹(t) = f(1-t). This traces the same path backward. The product f · f⁻¹ goes out along f and back along f⁻¹, which is homotopic to the constant loop. We've now verified all four group axioms for π₁(X, x₀): Closure: [f] · [g] = [f · g] is well-defined Associativity: ([f] · [g]) · [h] = [f] · ([g] · [h]) Identity: [e] · [f] = [f] = [f] · [e] Inverses: [f] · [f⁻¹] = [e] = [f⁻¹] · [f]

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