Circle Winding Numbers

Algebraic Topology · Axiom Academy

EXAMPLE Circle Winding Numbers Understanding how loops on S¹ correspond to integers and how composition equals addition Excellent work! You've completed this example. Here's what we learned: Winding numbers are integers: Each loop on the circle S¹ can be characterized by how many times it winds around the circle, giving a correspondence π₁(S¹) ≅ ℤ. Lifting to the universal cover: By viewing S¹ as ℝ/ℤ, we can lift loops to paths in ℝ, and the winding number equals the change in the lifted path's endpoint. Composition equals addition: When you compose two loops (first go around n times, then m times), the resulting winding number is n + m, making π₁(S¹) a group under composition. The fundamental group structure: This shows π₁(S¹) ≅ ℤ with the group operation being addition, where the generator is a loop that winds once counterclockwise. This fundamental example is the building block for computing fundamental groups of more complex spaces!

This is the written version of the interactive lesson above. See the full Algebraic Topology course.