The Universal Covering Space

Algebraic Topology · Axiom Academy

LESSON The Universal Covering Space The unique simply connected covering that covers all others 1. Definition of Universal Cover A covering space is called universal if it is simply connected, meaning its fundamental group is trivial. Not all spaces have universal covers, but nice spaces do. A space B has a universal covering if and only if it is path-connected, locally path-connected, and semi-locally simply connected. The condition "semi-locally simply connected" means: every point has a neighborhood U such that loops in U are contractible in B (not necessarily in U). Let's examine the universal covers of familiar spaces: Circle S¹: The universal cover is ℝ with projection p(t) = e^(2πit) Torus T²: The universal cover is ℝ² with projection identifying (x,y) ~ (x+m, y+n) Figure-eight: The universal cover is an infinite 4-valent tree Sphere S²: Already simply connected, so it's its own universal cover The universal cover is truly "universal" because it covers every other covering space. If p: E → B is the universal cover and q: F → B is any other connected covering, then there exists a covering map φ: E → F such that q ∘ φ = p.

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