Fundamental Group of the Circle

Algebraic Topology · Axiom Academy

LESSON Fundamental Group of the Circle Computing π₁(S¹) ≅ ℤ: a fundamental result in algebraic topology Loops on the circle can be classified by how many times they "wind around" the circle: The constant loop winds 0 times (trivial element) A counterclockwise loop winds +1 time A clockwise loop winds -1 time Multiple winds are possible: +2, -3, etc. This integer count is the winding number, and it's preserved under homotopy! 2. The Covering Space Argument The key insight uses the universal covering space p: ℝ → S¹ given by p(t) = e^(2πit): Homotopic loops lift to paths with the same endpoint, so the winding number is a well-defined homotopy invariant. The winding number gives us a homomorphism Φ: π₁(S¹, 1) → ℤ: Well-defined: Homotopic loops have the same winding number Homomorphism: Concatenating loops adds winding numbers Surjective: For each n ∈ ℤ, the loop that winds n times represents an element Injective: Only the constant loop has winding number 0 Therefore, Φ is an isomorphism! Since ℤ is cyclic, π₁(S¹) is generated by a single element: the loop ω that winds once counterclockwise around the circle. This gives us complete algebraic control over the topology of the circle!

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