Covering Spaces - Main Theorem

The classification theorem for covering spaces establishes the fundamental correspondence between subgroups of the fundamental group and covering spaces.

Theorem

(Classification of Covering Spaces) Let $X$ be path-connected, locally path-connected, and semi-locally simply connected, with basepoint $x_0$ . Then there is a bijection between:

Conjugacy classes of subgroups $H \subseteq \pi_1(X, x_0)$
Isomorphism classes of path-connected covering spaces $p : (\tilde{X}, \tilde{x}_0) \to (X, x_0)$

The correspondence assigns to $H$ the covering space $X_H$ with $p_*(\pi_1(X_H, \tilde{x}_0)) = H$ .

This theorem reveals the algebraic structure underlying the topology of covering spaces. The fundamental group encodes all possible ways to "cover" the space.

Theorem

(Universal Covering Space) Under the same hypotheses, there exists a unique (up to isomorphism) simply connected covering space $p : \tilde{X} \to X$ , called the universal covering space. It has the following properties:

$\pi_1(\tilde{X}) = \{e\}$ (simply connected)
$\text{Deck}(\tilde{X}/X) \cong \pi_1(X, x_0)$
Every other path-connected covering factors uniquely through $\tilde{X}$

The universal covering is "universal" in the categorical sense: it maps to every other covering space via covering maps.

Proof

Sketch of existence: Construct $\tilde{X}$ as the set of homotopy classes of paths in $X$ starting at $x_0$ . The projection $p : \tilde{X} \to X$ sends $[\gamma]$ to $\gamma(1)$ . The topology on $\tilde{X}$ is generated by path-lifting: for open $U \subseteq X$ and a path class $[\gamma]$ ending in $U$ , we take all extensions $[\gamma * \delta]$ where $\delta$ stays in $U$ .

This space is simply connected: any loop in $\tilde{X}$ based at the constant path $[e_{x_0}]$ corresponds to a path in $X$ from $x_0$ to $x_0$ , which defines an element of $\pi_1(X, x_0)$ . The loop in $\tilde{X}$ is nullhomotopic if and only if the corresponding element is trivial.

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Example

For $X = S^1$ with $\pi_1(S^1) = \mathbb{Z}$ , the universal cover is $\mathbb{R}$ with covering map $p(t) = e^{2\pi i t}$ . The subgroup $n\mathbb{Z} \subseteq \mathbb{Z}$ corresponds to the $n$ -fold cover $S^1 \to S^1$ , $z \mapsto z^n$ .

Theorem

(Galois Correspondence) Let $p : \tilde{X} \to X$ be the universal covering. Then:

Normal subgroups $N \triangleleft \pi_1(X)$ correspond to normal (regular) covering spaces
The quotient group $\pi_1(X) / N$ is isomorphic to the deck transformation group of the corresponding normal cover
Subgroup inclusions $H_1 \subseteq H_2$ correspond to covering maps $X_{H_1} \to X_{H_2}$

This establishes covering space theory as geometric Galois theory, where the fundamental group plays the role of the Galois group.

Remark

The classification theorem fails for spaces that are not semi-locally simply connected. The Hawaiian earring $\bigvee_{n=1}^\infty S^1_n$ (circles of radius $1/n$ joined at a point) has a complicated fundamental group and its covering space theory is much more subtle.

The correspondence between algebra and geometry provided by this theorem makes many computational problems tractable by translating them into group theory.