Covering Space - Universal Covers

Universal Covers

A connected covering space is a universal cover if it is simply connected. The name universal cover comes from the following important property: if the mapping q: D → X is a universal cover of the space X and the mapping p: C → X is any cover of the space X where the covering space C is connected, then there exists a covering map f : D → C such that p f= q. This can be phrased as

The universal cover of the space X covers all connected covers of the space X.

The map f is unique in the following sense: if we fix a point x in the space X and a point d in the space D with q(d) = x and a point c in the space C with p(c) = x, then there exists a unique covering map f: D → C such that p f= q and f(d) = c.

If the space X has a universal cover then that universal cover is essentially unique: if the mappings q₁ : D₁ → X and q₂: D₂ → X are two universal covers of the space X then there exists a homeomorphism f: D₁ → D₂ such that q₂ f = q₁.

The space X has a universal cover if it is connected, locally path-connected and semi-locally simply connected. The universal cover of the space X can be constructed as a certain space of paths in the space X.

The example R → S1 given above is a universal cover. The map S3 → SO(3) from unit quaternions to rotations of 3D space described in quaternions and spatial rotation is also a universal cover.

If the space X carries some additional structure, then its universal cover normally inherits that structure:

• if the space X is a manifold, then so is its universal cover D
• if the space X is a Riemann surface, then so is its universal cover D, and p is a holomorphic map
• if the space X is a Lorentzian manifold, then so is its universal cover. Furthermore, suppose the subset p−1(U) is a disjoint union of open sets each of which is diffeomorphic with U by the mapping p. If the space X contains a closed timelike curve (CTC), then the space X is timelike multiply connected (no CTC can be timelike homotopic to a point, as that point would not be causally well-behaved), its universal (diffeomorphic) cover is timelike simply connected (it does not contain a CTC).
• if X is a Lie group (as in the two examples above), then so is its universal cover D, and the mapping p is a homomorphism of Lie groups. In this case the universal cover is also called the universal covering group. This has particular application to representation theory and quantum mechanics, since ordinary representations of the universal covering group (D) are projective representations of the original (classical) group (X).

The universal cover first arose in the theory of analytic functions as the natural domain of an analytic continuation.