Section 3.3 Universal Coverings

Definition 3.11.

A space \(X\) is semilocally simply connected if every point \(x \in X\) has a neighbourhood \(U\) such that the inclusion \(i\colon U\hookrightarrow X\) induces the trivial homomorphism

\begin{equation*} i_*\colon \pi_1(U,x)\longrightarrow \pi_1(X,x). \end{equation*}

Equivalently, every loop in \(U\) based at \(x\) is null-homotopic in \(X\) (not necessarily in \(U\)). If \(X\) is locally path-connected, then \(U\) may be chosen path-connected; in that case every loop in \(U\text{,}\) regardless of basepoint, is null-homotopic in \(X\text{.}\)

Definition 3.12.

A covering \(p \colon \tilde{X} \to X\) of a connected, locally path-connected space is called a universal covering if \(\tilde{X}\) is simply connected. By Corollary 3.9, such a covering is unique up to basepoint-preserving isomorphism.

Example 3.13.

The map \(\mathbb{R} \to S^1\text{,}\) \(x \mapsto e^{2\pi ix}\text{,}\) is the universal cover of \(S^1\) (since \(\mathbb{R}\) is simply connected by Proposition 2.15).

Nothing in Definition 3.12 explains the word “universal”: simple connectivity is a condition on \(\tilde{X}\) alone, with no reference to the other coverings of \(X\text{.}\) The next proposition supplies the missing explanation. A universal covering covers every other connected covering of \(X\text{,}\) and does so in exactly one way once basepoints are chosen; it therefore sits on top of the whole tower of coverings of \(X\text{.}\)

Proposition 3.14. Universal property of the universal covering.

Let \(X\) be connected and locally path-connected, let \(p \colon \tilde{X} \to X\) be a universal covering with \(p(\tilde{x}_0) = x_0\text{,}\) and let \(q \colon Y \to X\) be any covering with \(Y\) connected and \(q(y_0) = x_0\text{.}\) Then there is a unique map \(r \colon \tilde{X} \to Y\) with

\begin{equation*} q \circ r = p \qquad\text{and}\qquad r(\tilde{x}_0) = y_0, \end{equation*}

and \(r\) is itself a covering map. In particular \(\tilde{X}\) is a universal covering of \(Y\) as well.

Proof.

Being a covering of a locally path-connected space, \(\tilde{X}\) is locally path-connected, and it is connected because it is simply connected. Since \(p_*\,\pi_1(\tilde{X}, \tilde{x}_0)\) is trivial, it is certainly contained in \(q_*\,\pi_1(Y, y_0)\text{,}\) so Proposition 3.8 applied to \(p\) and the covering \(q\) produces a unique lift \(r\) with \(r(\tilde{x}_0) = y_0\) and \(q \circ r = p\text{.}\)

Since \(X\) is locally path-connected, it is locally connected, and Proposition 3.3 shows that the morphism \(r\) is a covering map. Moreover, \(Y\) is locally path-connected because it is a covering of \(X\text{.}\) Since \(Y\) is connected and \(\tilde X\) is simply connected, \(r\) is therefore a universal covering of \(Y\text{.}\)

Proposition 3.15.

A connected, locally path-connected space \(X\) has a universal cover if and only if it is semilocally simply connected.

Proof.

Proof sketch. If \(p\colon\tilde X\to X\) is a universal covering, choose an evenly covered path-connected neighbourhood \(U\) of \(x\in X\text{.}\) Every loop in \(U\) lifts to a loop in a single sheet over \(U\text{.}\) The lifted loop is null-homotopic because \(\tilde X\) is simply connected, so the original loop is null-homotopic in \(X\text{.}\)

Conversely, fix \(x_0\in X\) and let \(\tilde X\) be the set of endpoint-preserving homotopy classes of paths in \(X\) starting at \(x_0\text{,}\) with \(p([\alpha])=\alpha(1)\text{.}\) Give \(\tilde X\) the topology in which a basic neighbourhood of \([\alpha]\) is obtained by appending to \(\alpha\) paths lying in a small path-connected open set \(U\) whose loops are null-homotopic in \(X\text{.}\) Local path-connectedness and semilocal simple connectivity provide a basis of such sets \(U\text{,}\) and over each one the endpoint map is a disjoint union of homeomorphisms onto \(U\text{.}\) Thus \(p\) is a covering map. The lift of a path \(\alpha\) starting at \(x_0\) ends at \([\alpha]\text{,}\) from which one checks that \(\tilde X\) is path-connected and simply connected. Hence \(p\) is a universal covering. This is the standard path-space construction; see Fulton, Algebraic Topology: A First Course, Theorem 13.20.

Exercise 3.16.

Suppose \(X\) is locally path-connected and semilocally simply connected. If \(p \colon Y \to X\) and \(q \colon Z \to Y\) are covering maps, show that \(p \circ q \colon Z \to X\) is a covering map.

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