Coverings of the Torus

Algebraic Topology · Axiom Academy

EXAMPLE Coverings of the Torus Exploring covering spaces via the correspondence with subgroups of ℤ² Excellent work! You've explored the rich structure of torus coverings. Here's what we learned: Universal Cover: The plane ℝ² is the universal covering space of the torus T² = S¹ × S¹ Cylinder Covers: ℝ × S¹ → T² are infinite-sheeted covers corresponding to subgroups ℤ × 0 and 0 × ℤ Finite Covers: Sublattices of ℤ² give finite-sheeted torus-to-torus covers, including n × m-fold covers Subgroup Correspondence: Covering spaces of T² bijectively correspond to subgroups of π₁(T²) = ℤ² Rich Structure: Unlike S¹, the torus has infinitely many non-isomorphic finite covers of each degree The torus provides a fascinating laboratory for understanding covering space theory!

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