A universal cover factors through every connected cover, and its deck group is the fundamental group.Suggest a slogan
Let \(X\) be path-connected, locally path-connected and semilocally simply connected. Then \(X\) has a universal cover \(p\colon\hat X \to X\) [@Hat02]. If \(q\colon C\to X\) is a covering map with \(C\) path-connected, then there is a covering map \(\tilde p\colon \hat{X} \to C\) with \(q\circ\tilde p = p\) [@Hat02]:
In particular, \(\hat X\) is unique up to isomorphism of covering spaces.
The deck transformation group of the universal cover is isomorphic to \(\pi_1(X)\). More generally, for a covering map \(q\colon C\to X\) with \(C\) path-connected and \(H\coloneqq q_*\pi_1(C)\leq\pi_1(X)\), the deck transformation group of \(q\) is isomorphic to $N_{\pi_1(X)}(H)/H$ [@Hat02].