Covering Spaces - Key Properties

Covering spaces possess remarkable lifting properties that make them indispensable tools for computing fundamental groups and analyzing continuous maps.

Theorem

(Path Lifting Property) Let $p : (\tilde{X}, \tilde{x}_0) \to (X, x_0)$ be a covering space. For any path $\gamma : [0,1] \to X$ with $\gamma(0) = x_0$ , there exists a unique lift $\tilde{\gamma} : [0,1] \to \tilde{X}$ such that $p \circ \tilde{\gamma} = \gamma$ and $\tilde{\gamma}(0) = \tilde{x}_0$ .

The uniqueness is crucial: once we specify a starting point in the fiber, the lift is completely determined. This property reflects the local homeomorphism nature of covering maps.

Theorem

(Homotopy Lifting Property) Let $p : \tilde{X} \to X$ be a covering space and $H : Y \times [0,1] \to X$ a homotopy with $H(y,0) = h_0(y)$ for some map $h_0 : Y \to X$ . If $\tilde{h}_0 : Y \to \tilde{X}$ is a lift of $h_0$ , then there exists a unique lift $\tilde{H} : Y \times [0,1] \to \tilde{X}$ of $H$ with $\tilde{H}(y,0) = \tilde{h}_0(y)$ .

This property says that homotopies lift uniquely once we specify the initial lift. Combined with path lifting, it implies that homotopic paths have homotopic lifts.

Definition

A covering map $p : \tilde{X} \to X$ induces a monomorphism on fundamental groups: $p_* : \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0)$ The image $p_*(\pi_1(\tilde{X}, \tilde{x}_0))$ is a subgroup of $\pi_1(X, x_0)$ .

Theorem

If $\tilde{X}$ is path-connected, the induced homomorphism $p_* : \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0)$ is injective. Moreover, a loop $\gamma$ in $X$ based at $x_0$ lifts to a loop in $\tilde{X}$ based at $\tilde{x}_0$ if and only if $[\gamma] \in p_*(\pi_1(\tilde{X}, \tilde{x}_0))$ .

This establishes the fundamental correspondence: subgroups of $\pi_1(X, x_0)$ correspond to path-connected covering spaces of $X$ .

Example

For the covering $p : \mathbb{R} \to S^1$ given by $p(t) = e^{2\pi i t}$ , we have $\pi_1(\mathbb{R}) = \{e\}$ and $\pi_1(S^1) = \mathbb{Z}$ . The induced map $p_* : \{e\} \to \mathbb{Z}$ is the trivial subgroup. Any loop in $S^1$ lifts to a path in $\mathbb{R}$ , but this lift is never closed: it "unwraps" the loop.

Definition

A covering space $p : \tilde{X} \to X$ is normal (or regular) if for any two points $\tilde{x}_1, \tilde{x}_2 \in p^{-1}(x_0)$ , there exists a deck transformation (covering automorphism) taking $\tilde{x}_1$ to $\tilde{x}_2$ .

Remark

Normal covering spaces correspond to normal subgroups of $\pi_1(X, x_0)$ . The group of deck transformations is isomorphic to the quotient $\pi_1(X, x_0) / p_*(\pi_1(\tilde{X}, \tilde{x}_0))$ . This connection makes covering space theory a geometric manifestation of group theory.

The lifting properties make covering spaces computational tools: to understand maps and homotopies in $X$ , we lift them to $\tilde{X}$ where the topology may be simpler.