Covering Spaces - Examples and Constructions

Understanding covering spaces requires examining key examples and construction techniques that reveal the deep connections between topology and algebra.

Example

Universal cover of $S^1$ : The covering $p : \mathbb{R} \to S^1$ given by $p(t) = e^{2\pi i t}$ is the universal covering of $S^1$ . It is simply connected and covers all other connected coverings of $S^1$ . The deck transformation group is $\mathbb{Z}$ , acting by translations $t \mapsto t + n$ .

The universal covering, when it exists, is the unique (up to isomorphism) simply connected covering space that dominates all other connected covers.

Example

Covering of the torus: The map $p : \mathbb{R}^2 \to T^2 = S^1 \times S^1$ given by $p(x,y) = (e^{2\pi i x}, e^{2\pi i y})$ is the universal covering of the torus. The deck transformation group is $\mathbb{Z}^2$ , acting by lattice translations. This covering "unwraps" the torus into the plane.

Definition

Let $p : \tilde{X} \to X$ be a covering space. A deck transformation (or covering transformation) is a homeomorphism $\phi : \tilde{X} \to \tilde{X}$ such that $p \circ \phi = p$ . The set of all deck transformations forms a group $\text{Deck}(\tilde{X}/X)$ under composition.

Theorem

If $p : \tilde{X} \to X$ is a covering with $\tilde{X}$ path-connected and $X$ path-connected and locally path-connected, then: $\text{Deck}(\tilde{X}/X) \cong \pi_1(X, x_0) / p_*(\pi_1(\tilde{X}, \tilde{x}_0))$

This isomorphism is the heart of the Galois correspondence in covering space theory. For the universal covering (where $\pi_1(\tilde{X}) = \{e\}$ ), we get $\text{Deck}(\tilde{X}/X) \cong \pi_1(X)$ .

Example

The $n$ -fold covers of $S^1$ : For each $n \geq 1$ , the map $p_n : S^1 \to S^1$ given by $p_n(z) = z^n$ is an $n$ -fold covering. These are all the connected covers of $S^1$ up to equivalence, corresponding to the subgroups $n\mathbb{Z} \subseteq \mathbb{Z} = \pi_1(S^1)$ .

Definition

A space $X$ has a universal covering space if there exists a simply connected covering $p : \tilde{X} \to X$ . A sufficient condition is that $X$ be path-connected, locally path-connected, and semi-locally simply connected.

Semi-local simple connectivity requires that each point has a neighborhood whose loops are contractible in $X$ . This rules out pathological spaces like the Hawaiian earring.

Example

Covering of punctured plane: The space $\mathbb{R}^2 \setminus \{0\}$ deformation retracts to $S^1$ , so its universal cover is $\mathbb{R}$ . The covering map can be constructed explicitly using the exponential map in polar coordinates.

Remark

Covering spaces provide a geometric interpretation of quotient groups: if $H \subseteq G = \pi_1(X)$ is a subgroup, there exists a covering $p : X_H \to X$ with $\pi_1(X_H) = H$ and $p_*$ the inclusion map. This establishes a bijection between subgroups of $\pi_1(X)$ and isomorphism classes of path-connected coverings of $X$ .

The construction of covering spaces from group-theoretic data makes abstract algebra concrete through geometric visualization.