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Section 3.2 The Lifting Criterion

We consider the problem of classifying coverings of a fixed space \(X\text{.}\) The key tool is the following lifting criterion. Recall from Proposition 2.12 that for a covering \(p \colon \tilde{X} \to X\) with \(p(\tilde{x}_0) = x_0\text{,}\) the map \(p_*\) is injective. Thus a connected covering with a chosen point over \(x_0\) determines a subgroup \(\operatorname{im}(p_*) \subseteq \pi_1(X, x_0)\text{.}\) Under the local hypotheses introduced below, this subgroup is a complete invariant up to basepoint-preserving isomorphism. The following example computes the subgroup for the coverings of \(S^1\) we already know.

Example 3.6.

Recall the covering \(p \colon S^1 \to S^1\) given by \(z \mapsto z^n\text{,}\) for \(n \geq 1\text{.}\) Let \(\gamma \colon I \to S^1\text{,}\) \(\gamma(t) = e^{2\pi i t}\text{,}\) be a generator of the fundamental group upstairs, and write \(\omega\) for the same map thought of as generating the fundamental group downstairs. Then
\begin{equation*} (p \circ \gamma)(t) = e^{2\pi i n t}, \end{equation*}
so \(p_*[\gamma] = [\omega]^n\) by Remark 2.17. Under the isomorphism \(\pi_1(S^1, 1) \cong \mathbb{Z}\) of Theorem 2.16 this says
\begin{equation*} p_*\,\pi_1(S^1, 1) = n\mathbb{Z} \subseteq \mathbb{Z}. \end{equation*}
Varying \(n\text{,}\) and taking the covering \(\mathbb{R} \to S^1\) of Theorem 2.16 for the trivial subgroup, we realise every subgroup of \(\pi_1(S^1, 1)\) as the image of \(p_*\) for some covering of \(S^1\text{.}\) This is no accident: see Proposition 3.17.

1.
If \(\tilde{\alpha}\) is a loop at \(\tilde{x}_0\text{,}\) then \([\alpha] = [p \circ \tilde{\alpha}] = p_*[\tilde{\alpha}] \in H\text{.}\)
Conversely, suppose \([\alpha] = p_*[\tilde{\gamma}]\) for some loop \(\tilde{\gamma}\) at \(\tilde{x}_0\text{,}\) so that \(\alpha \simeq_p p \circ \tilde{\gamma}\) via a path-homotopy \(H'\text{.}\) By Corollary 2.10 we may lift \(H'\) to a path-homotopy \(\tilde{H}'\) starting at \(\tilde{x}_0\text{.}\) Its two ends are lifts of \(\alpha\) and of \(p \circ \tilde{\gamma}\) starting at \(\tilde{x}_0\text{,}\) hence are \(\tilde{\alpha}\) and \(\tilde{\gamma}\) by uniqueness of lifts (Lemma 2.8). A path-homotopy fixes endpoints, so \(\tilde{\alpha}(1) = \tilde{\gamma}(1) = \tilde{x}_0\) and \(\tilde{\alpha}\) is a loop.
(2) Let \(\tilde{\delta}\) be the lift of \(\bar{\alpha}\) starting at \(\tilde{\beta}(1)\text{.}\) Then \(\tilde{\beta} * \tilde{\delta}\) is a lift of \(\beta * \bar{\alpha}\) starting at \(\tilde{x}_0\text{,}\) so it is the lift, again by Lemma 2.8. By part (1),
\begin{equation*} [\beta * \bar{\alpha}] \in H \iff \tilde{\beta} * \tilde{\delta} \text{ is a loop} \iff \tilde{\delta}(1) = \tilde{x}_0 . \end{equation*}
Now \(\overline{\tilde{\delta}}\) is a lift of \(\alpha\) running from \(\tilde{\delta}(1)\) to \(\tilde{\beta}(1)\text{.}\) So if \(\tilde{\delta}(1) = \tilde{x}_0\text{,}\) then \(\overline{\tilde{\delta}} = \tilde{\alpha}\) by uniqueness and \(\tilde{\alpha}(1) = \tilde{\beta}(1)\text{.}\) Conversely, if \(\tilde{\alpha}(1) = \tilde{\beta}(1)\text{,}\) then \(\overline{\tilde{\alpha}}\) is a lift of \(\bar{\alpha}\) starting at \(\tilde{\beta}(1)\text{,}\) so \(\tilde{\delta} = \overline{\tilde{\alpha}}\) and \(\tilde{\delta}(1) = \tilde{\alpha}(0) = \tilde{x}_0\text{.}\)

Proof.
For necessity, if \(\tilde{f}\) exists then \(f_* = p_* \circ \tilde{f}_*\text{,}\) so \(\operatorname{im}(f_*) \subseteq \operatorname{im}(p_*)\text{.}\)
For sufficiency, note first that \(Y\) is path-connected. For \(y \in Y\) choose a path \(\alpha \colon I \to Y\) from \(y_0\) to \(y\) and set
\begin{equation*} \tilde{f}(y) := \text{endpoint of the lift of } f \circ \alpha \text{ starting at } \tilde{x}_0. \end{equation*}
To see that this is well-defined, if \(\alpha'\) is another path from \(y_0\) to \(y\text{,}\) then \(f \circ (\alpha' * \bar{\alpha})\) is a loop at \(x_0\text{.}\) By hypothesis its class is in \(\operatorname{im}(p_*)\text{,}\) so by Proposition 3.7(2) the lifts of \(f \circ \alpha\) and \(f \circ \alpha'\) end at the same point.
To see that it is continuous, let \(U\) be an evenly covered neighbourhood of \(f(y)\) and choose a path-connected open neighbourhood \(W\) of \(y\) with \(f(W) \subseteq U\text{.}\) Under a trivialization
\begin{equation*} p^{-1}(U) \cong U \times F, \end{equation*}
where \(F\) is discrete, write \(\tilde{f}(y)=(f(y),c)\) for some \(c\in F\text{.}\) For any \(w\in W\text{,}\) choose a path \(\gamma\) in \(W\) from \(y\) to \(w\text{.}\) The lift of \(f\circ(\alpha*\gamma)\) extends the lift of \(f\circ\alpha\text{,}\) so its final segment is the lift of \(f\circ\gamma\) starting at \(\tilde f(y)\text{.}\) In the chosen trivialization this lift is
\begin{equation*} t\longmapsto \bigl(f(\gamma(t)),c\bigr). \end{equation*}
It follows that \(\tilde f(w)=(f(w),c)\) for every \(w\in W\text{.}\) Thus
\begin{equation*} \tilde f|_W=(p|_{U\times\{c\}})^{-1}\circ f|_W, \end{equation*}
which is continuous. Hence \(\tilde f\) is continuous near every point of \(Y\text{.}\)
Uniqueness follows from Lemma 2.8.

Proof.
Necessity is clear. For sufficiency, if the images agree then Proposition 3.8 gives maps \(\phi \colon \tilde{X}_1 \to \tilde{X}_2\) and \(\psi \colon \tilde{X}_2 \to \tilde{X}_1\) each preserving basepoints and compatible with the covering maps. By Lemma 2.8 they are mutual inverses.

Exercise 3.10.

Under the hypotheses of Corollary 3.9, choose \(\tilde{x}_i \in p_i^{-1}(x_0)\) for \(i=1,2\text{.}\) The connected coverings \(p_1\) and \(p_2\) are isomorphic in \(\operatorname{Cov}(X)\text{,}\) in the sense of Definition 3.4, by an isomorphism not necessarily carrying \(\tilde{x}_1\) to \(\tilde{x}_2\text{,}\) if and only if the subgroups
\begin{equation*} (p_i)_*\,\pi_1(\tilde{X}_i,\tilde{x}_i) \subseteq \pi_1(X,x_0) \end{equation*}
are conjugate. In particular, the conjugacy class is independent of the chosen point over \(x_0\text{.}\)
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