Coverings of the Circle

Algebraic Topology · Axiom Academy

EXAMPLE Coverings of the Circle Classifying all covering spaces of S¹ using fundamental group theory Excellent work! You've completed the classification of covers of S¹. Here's what we learned: Universal Cover: ℝ → S¹ via p(t) = e^(2πit) is the simply connected universal covering space n-fold Covers: For each n ≥ 1, we have S¹ → S¹ via z ↦ z^n, corresponding to subgroups nℤ ⊆ ℤ Galois Correspondence: There is a bijection between connected covering spaces and subgroups of π₁(S¹) = ℤ Deck Transformations: The group of deck transformations of ℝ → S¹ is isomorphic to ℤ, generated by t ↦ t + 1 This fundamental example illustrates the power of covering space theory in algebraic topology!

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