Classification of Covering Spaces
Covers of a space are subgroups of its fundamental group, and nothing less.
The idea
The classification theorem for covering spaces matches the connected covering spaces of a space $X$ with the subgroups of its fundamental group, so that finding every cover of $X$ becomes an algebra problem.
A covering space is a map $p \colon \tilde{X} \to X$ that lays evenly stacked sheets over every small enough open set $U \subseteq X$ (Covering Spaces). Each connected cover records a subgroup. Take a loop in $X$ at the basepoint $x_{0}$ and lift it, starting at a chosen point $\tilde{x}_{0}$ of the fibre $p^{-1}(x_{0})$: the lifted path either closes up into a loop or ends at another point of the fibre. The classes of loops whose lifts close up form the subgroup $H = p_{*}\pi_{1}(\tilde{X}, \tilde{x}_{0})$ of $\pi_{1}(X, x_{0})$.
Theorem (Classification of covering spaces).
Let $X$ be path-connected, locally path-connected, and semilocally simply connected, with basepoint $x_{0}$. Sending a basepointed connected cover $p \colon (\tilde{X}, \tilde{x}_{0}) \to (X, x_{0})$ to the subgroup $p_{*}\pi_{1}(\tilde{X}, \tilde{x}_{0})$ is a bijection between basepointed connected covers of $X$ and subgroups of $\pi_{1}(X, x_{0})$. Forgetting basepoints, connected covers correspond to conjugacy classes of subgroups.
Every cell complex satisfies the hypotheses. Under this correspondence the number of sheets is the index of $H$, the trivial subgroup gives the simply connected universal cover, and a normal subgroup gives a cover whose symmetries permute the sheets transitively, with the quotient $\pi_{1}(X, x_{0})/H$ as the group of those symmetries. The correspondence has the same shape as Galois theory, and borrows its name.
Ways to work on it
- Walkthrough. The Galois correspondence, the lifting criterion, sheets as index, universal covers and deck groups.
- Proof. See why the number of sheets is the index — lift loops and match the fibre with the cosets.
- Practice. Count sheets from a homomorphism, find a deck group, and identify a universal cover.
- Hardest. Count the sheets of a cover from its subgroup, and meet a cover with no symmetry at all.
Not sure where to start? Take the ten-question placement test.