Classification of Covering Spaces

Algebraic Topology · Axiom Academy

LESSON Classification of Covering Spaces Covering spaces correspond to subgroups of the fundamental group 1. The Fundamental Correspondence For a path-connected, locally path-connected, and semi-locally simply connected space B with basepoint b₀, there is a bijection: This means every covering space determines a subgroup, and every subgroup determines a covering space (up to isomorphism). The degree of a covering (number of sheets) equals the index of the corresponding subgroup in the fundamental group. This gives us a way to count covering spaces: an n-fold covering corresponds to a subgroup of index n. The circle S¹ has fundamental group π₁(S¹) ≅ ℤ. The subgroups of ℤ are exactly nℤ for n ≥ 0, where 0ℤ = 0 . Trivial subgroup 0 : Universal cover ℝ → S¹ (infinite-sheeted) Subgroup nℤ: n-fold cover S¹ → S¹, z ↦ z^n (n sheets) Since ℤ has one subgroup of each index n, the circle has exactly one n-fold connected covering for each n. 4. Reconstructing Coverings from Subgroups Given a subgroup H ≤ π₁(B, b₀), we can explicitly construct the corresponding covering space. The covering space E_H consists of homotopy classes of paths in B, with the topology making the projection continuous. This construction shows the correspondence is not just a counting result, but provides an explicit way to build any covering from group-theoretic data.

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