Universal cover - Knowlpedia

Let $X$ be a topological space. A universal cover of $X$ is a covering map

p:\widetilde X\to X

such that $\widetilde X$ is connected and simply connected. Thus the total space is a connected simply connected space that covers $X$ .

Existence and uniqueness

If $X$ is connected, locally path-connected, and semilocally simply connected, then a universal cover exists. It is unique up to a homeomorphism over $X$ . In particular, every connected smooth manifold has a universal cover, since smooth manifolds satisfy these local hypotheses.

Deck transformations

A deck transformation is a homeomorphism $D:\widetilde X\to\widetilde X$ with $p\circ D=p$ . After choosing a basepoint in $X$ and a lift, the deck transformation group of the universal cover is identified with the fundamental group $\pi_1(X)$ , up to the left/right action convention used for lifting paths. The basepoint choice changes this identification by the corresponding standard conjugacy ambiguity.

For a path-connected, locally path-connected, semilocally simply connected base, every connected covering is obtained from the universal cover by a subgroup of $\pi_1(X)$ , with the usual conjugacy ambiguity when no lifted basepoint is chosen.

Examples

The identity map is the universal cover of a simply connected space. The exponential covering $\mathbb R\to S^1$ , $t\mapsto e^{2\pi i t}$ , is the universal cover of the circle; its deck group is isomorphic to $\mathbb Z$ .

References

Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002, Chapter 1, §1.3, “Covering Spaces,” especially pp. 60–73. Author-hosted PDF.