Section 3.5 Coverings as \(\pi_1\)-Sets (Optional)

The classification above is stated in terms of subgroups. There is an equivalent formulation, closer to Galois theory, in which a covering is remembered by its fibre together with the action of \(\pi_1(X, x_0)\) on it by lifting loops. Definition 2.13 is the special case of this action applied to one point.

Definition 3.20.

Let \(p \colon \tilde{X} \to X\) be a covering with \(x_0 \in X\text{,}\) and let \(F = p^{-1}(x_0)\) be the fibre. For \(\tilde{x} \in F\) and \([\gamma] \in \pi_1(X, x_0)\text{,}\) set

\begin{equation*} \tilde{x} \cdot [\gamma] := \tilde{\gamma}(1), \qquad \tilde{\gamma} = \text{the lift of } \gamma \text{ starting at } \tilde{x}. \end{equation*}

This is well-defined by Corollary 2.10, and is a right action of \(\pi_1(X, x_0)\) on \(F\text{,}\) called the monodromy action.

Exercise 3.21.

Check that \(\tilde{x} \cdot [c_{x_0}] = \tilde{x}\) and \(\tilde{x} \cdot ([\gamma] * [\delta]) = (\tilde{x} \cdot [\gamma]) \cdot [\delta]\text{,}\) so that Definition 3.20 really is an action.

Proposition 3.22.

Let \(X\) be path-connected and let \(p \colon \tilde{X} \to X\) be a covering with \(\tilde{X}\) path-connected and \(p(\tilde{x}_0) = x_0\text{.}\) Write \(H = p_*\,\pi_1(\tilde{X}, \tilde{x}_0)\text{.}\) Then the monodromy action on \(F = p^{-1}(x_0)\) is transitive, the stabiliser of \(\tilde{x}_0\) is \(H\text{,}\) and the map

\begin{equation*} H \backslash \pi_1(X, x_0) \longrightarrow F, \qquad H[\gamma] \longmapsto \tilde{x}_0 \cdot [\gamma] \end{equation*}

is a bijection. In particular the number of sheets of \(p\) equals the index \([\pi_1(X, x_0) : H]\text{.}\)

Proof.

Transitivity is the surjectivity statement of Proposition 2.14, which only used path-connectedness of \(\tilde{X}\text{.}\) The stabiliser of \(\tilde{x}_0\) consists of those \([\gamma]\) whose lift at \(\tilde{x}_0\) is a loop, which is exactly \(H\) by Proposition 3.7(1). The displayed map is therefore the orbit–stabiliser bijection, and the number of sheets is \(|F|\text{.}\)

Example 3.23.

The \(n\)-sheeted connected coverings of \(S^1\) correspond to the index-\(n\) subgroups of \(\pi_1(S^1) \cong \mathbb{Z}\text{,}\) of which there is exactly one, namely \(n\mathbb{Z}\text{.}\) This recovers the answer to Exercise 3.19: up to isomorphism the connected coverings of \(S^1\) are \(z \mapsto z^n\) for \(n \geq 1\text{,}\) together with \(\mathbb{R} \to S^1\text{.}\)

Proposition 3.24. Categorical classification.

Let \(X\) be connected, locally path-connected, and semilocally simply connected. The fibre functor

\begin{equation*} \Phi\colon\operatorname{Cov}(X)\longrightarrow \{\text{sets with a right }\pi_1(X,x_0)\text{-action}\}, \qquad p\longmapsto p^{-1}(x_0), \end{equation*}

is an equivalence of categories. Under this equivalence, connected coverings correspond to transitive actions and the universal cover corresponds to \(\pi_1(X,x_0)\) acting on itself on the right.

Proof.

A morphism of coverings restricts on fibres to an equivariant map, by uniqueness of path lifting. Conversely, let \(p\colon\tilde X\to X\) and \(q\colon Y\to X\) be coverings and let \(\varphi\colon p^{-1}(x_0)\to q^{-1}(x_0)\) be equivariant. Each component \(C\) of \(\tilde X\) is an open, path-connected covering of \(X\) and meets \(p^{-1}(x_0)\text{.}\) Choose \(\tilde x\in C\cap p^{-1}(x_0)\text{,}\) and let \(D\) be the component of \(Y\) containing \(\varphi(\tilde x)\text{.}\) Equivariance gives an inclusion of stabilisers

\begin{equation*} \operatorname{Stab}(\tilde x)\subseteq\operatorname{Stab}(\varphi(\tilde x)). \end{equation*}

By Proposition 3.7(1), these stabilisers are respectively

\begin{equation*} \operatorname{Stab}(\tilde x)=p_*\pi_1(C,\tilde x),\qquad \operatorname{Stab}(\varphi(\tilde x))=q_*\pi_1(D,\varphi(\tilde x)). \end{equation*}

The lifting criterion therefore gives a unique map \(C\to D\) over \(X\) taking \(\tilde x\) to \(\varphi(\tilde x)\text{.}\) Equivariance and uniqueness of path lifting show that its restriction to the fibre is \(\varphi\) on the orbit of \(\tilde x\text{.}\) Applying this construction on every component gives a unique morphism \(\tilde X\to Y\) inducing \(\varphi\text{.}\) Thus \(\Phi\) is fully faithful.

For essential surjectivity, decompose any right \(\pi_1(X,x_0)\)-set into orbits. After choosing a point in an orbit, it is isomorphic to \(H\backslash\pi_1(X,x_0)\) for its stabiliser \(H\text{.}\) By Proposition 3.17, Proposition 3.22, this orbit is the fibre of the connected covering \(X_H\to X\text{.}\) The disjoint union of these coverings realises the original action. Finally, Proposition 3.22 identifies connected coverings with transitive actions, and the universal cover, whose stabiliser is trivial, with the right regular action.

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