Covering Spaces Introduction

Covering spaces are a fundamental tool in algebraic topology, intimately connected with the fundamental group. They provide a geometric way to "unwind" loops in a space and lead to the classification of coverings via subgroups of the fundamental group.

Definition

Definition9.10Covering Space

A covering space of a topological space $X$ is a space $\tilde{X}$ together with a continuous surjection $p: \tilde{X} \to X$ (the covering map) such that every point $x \in X$ has an open neighborhood $U$ that is evenly covered: $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$ , each of which is mapped homeomorphically onto $U$ by $p$ .

The open sets in $p^{-1}(U)$ are called the sheets of the covering over $U$ . The set $p^{-1}(x)$ is called the fiber over $x$ .

ExampleCovering Spaces

$\mathbb{R} \to S^1$ : The map $p: \mathbb{R} \to S^1$ defined by $p(t) = e^{2\pi i t}$ is a covering map. Each point of $S^1$ has a neighborhood evenly covered by countably many intervals in $\mathbb{R}$ . The fiber over each point is $\mathbb{Z}$ .
$S^n \to \mathbb{R}P^n$ : The quotient map $p: S^n \to S^n/(x \sim -x) = \mathbb{R}P^n$ is a two-sheeted covering. Each fiber has exactly two points (a point and its antipode).
$\mathbb{R}^2 \to T^2$ : The map $(x, y) \mapsto (e^{2\pi i x}, e^{2\pi i y})$ is a covering of the torus with fiber $\mathbb{Z}^2$ .
Trivial covering: $X \times F \to X$ (projection, where $F$ is discrete) is a covering. Every covering is locally of this form.
Identity covering: $\operatorname{id}: X \to X$ is a covering (the trivial one-sheeted covering).

Properties of Covering Maps

Theorem9.4Basic Properties

Let $p: \tilde{X} \to X$ be a covering map.

$p$ is a local homeomorphism.
$p$ is an open map.
If $X$ is connected, all fibers $p^{-1}(x)$ have the same cardinality (the degree or number of sheets).
If $X$ is Hausdorff, so is $\tilde{X}$ . Similarly for second-countable, locally path-connected, etc.

Path Lifting

Theorem9.5Path Lifting Property

Let $p: \tilde{X} \to X$ be a covering map, $\gamma: [0, 1] \to X$ a path, and $\tilde{x}_0 \in p^{-1}(\gamma(0))$ . Then there exists a unique path $\tilde{\gamma}: [0, 1] \to \tilde{X}$ (called the lift of $\gamma$ ) with $\tilde{\gamma}(0) = \tilde{x}_0$ and $p \circ \tilde{\gamma} = \gamma$ .

Proof

Existence: Cover $\gamma([0, 1])$ by evenly covered neighborhoods. By compactness of $[0, 1]$ , finitely many suffice. Use the Lebesgue number lemma to partition $[0, 1]$ into subintervals, each mapping into an evenly covered neighborhood. Lift $\gamma$ inductively on each subinterval using the homeomorphic sheets.

Uniqueness: If $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ are two lifts with $\tilde{\gamma}_1(0) = \tilde{\gamma}_2(0)$ , the set $\{t : \tilde{\gamma}_1(t) = \tilde{\gamma}_2(t)\}$ is both open and closed in $[0, 1]$ (using the local homeomorphism property). Since $[0, 1]$ is connected and this set is nonempty, it equals $[0, 1]$ .

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Theorem9.6Homotopy Lifting Property

Let $p: \tilde{X} \to X$ be a covering map and $H: [0, 1] \times [0, 1] \to X$ a homotopy with $H(0, 0)$ lifted to $\tilde{x}_0$ . Then there exists a unique lift $\tilde{H}: [0, 1] \times [0, 1] \to \tilde{X}$ with $\tilde{H}(0, 0) = \tilde{x}_0$ and $p \circ \tilde{H} = H$ .

The Fundamental Group and Coverings

Theorem9.7Induced Homomorphism

If $p: (\tilde{X}, \tilde{x}_0) \to (X, x_0)$ is a covering map, then: $p_*: \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0)$ is an injective homomorphism. The image $p_*(\pi_1(\tilde{X}, \tilde{x}_0))$ is a subgroup of $\pi_1(X, x_0)$ .

Proof

$p_*$ is a homomorphism by functoriality. For injectivity: if $p_*([\tilde{\gamma}]) = [e]$ (the constant loop), then $p \circ \tilde{\gamma}$ is null-homotopic in $X$ via a homotopy $H$ . By the homotopy lifting property, $H$ lifts to $\tilde{H}$ with $\tilde{H}_0 = \tilde{\gamma}$ . Since $H_1$ is constant at $x_0$ , $\tilde{H}_1$ is a constant loop at $\tilde{x}_0$ (by uniqueness of lifting). So $[\tilde{\gamma}] = [e]$ in $\pi_1(\tilde{X}, \tilde{x}_0)$ .

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The Universal Cover

Definition9.11Universal Covering Space

A covering space $\tilde{X}$ of $X$ is a universal covering space if $\tilde{X}$ is simply connected (path-connected with trivial fundamental group).

Theorem9.8Existence of Universal Cover

If $X$ is connected, locally path-connected, and semi-locally simply connected, then $X$ has a universal covering space. The universal cover is unique up to isomorphism of covering spaces.

ExampleUniversal Covers

$\mathbb{R} \xrightarrow{p} S^1$ : $\mathbb{R}$ is the universal cover of $S^1$ .
$S^n \xrightarrow{p} \mathbb{R}P^n$ : $S^n$ (simply connected for $n \geq 2$ ) is the universal cover of $\mathbb{R}P^n$ .
$\mathbb{R}^2 \xrightarrow{p} T^2$ : $\mathbb{R}^2$ is the universal cover of the torus.
The universal cover of the figure-eight is an infinite tree (the Cayley graph of $F_2$ ).

RemarkClassification of Coverings

For a "nice" space $X$ (connected, locally path-connected, semi-locally simply connected), there is a bijection between:

Isomorphism classes of connected coverings $p: \tilde{X} \to X$ , and
Conjugacy classes of subgroups of $\pi_1(X, x_0)$ .

The covering corresponding to a subgroup $H \leq \pi_1(X)$ has $p_*(\pi_1(\tilde{X})) = H$ , and the number of sheets equals $[\pi_1(X) : H]$ . The universal cover corresponds to the trivial subgroup.