Definition 2.7.
Let \(p \colon \tilde{X} \to X\) be a covering map. A lift of a map \(f \colon Y \to X\) is a map \(\tilde{f} \colon Y \to \tilde{X}\) such that \(p \circ \tilde{f} = f\text{,}\) i.e. the diagram
commutes.
Section 2.2 Lifting Properties
Lemma 2.8. Uniqueness of lifts.
Let \(p \colon \tilde{X} \to X\) be a covering map and \(Y\) connected. If \(\tilde{f}_1, \tilde{f}_2 \colon Y \to \tilde{X}\) are both lifts of \(f \colon Y \to X\) and agree at some point \(y \in Y\text{,}\) then \(\tilde{f}_1 = \tilde{f}_2\text{.}\)
Proof.
It suffices to show that the set \(A = \{y \in Y \colon \tilde{f}_1(y) = \tilde{f}_2(y)\}\) is both open and closed (connectedness then forces \(A = Y\) since \(A \ne \emptyset\) by hypothesis).
Let \(y \in Y\) and take a neighbourhood \(U\) of \(f(y)\) evenly covered by \(p\text{,}\) with sheets \(p^{-1}(U) = \bigsqcup_\alpha V_\alpha\text{.}\) By continuity there is an open \(W \ni y\) with \(\tilde{f}_i(W) \subseteq V_{\alpha_i}\) for some sheet \(V_{\alpha_i}\text{.}\)
If \(y \in A\text{,}\) then \(\tilde{f}_1(y) = \tilde{f}_2(y)\) lies in a single sheet \(V_\alpha\text{,}\) so both maps send \(W\) into \(V_\alpha\text{;}\) since \(p|_{V_\alpha}\) is a homeomorphism and \(p \circ \tilde{f}_1 = p \circ \tilde{f}_2
= f\) on \(W\text{,}\) we get \(\tilde{f}_1 = \tilde{f}_2\) on \(W\text{.}\) Hence \(W \subseteq A\text{,}\) so \(A\) is open.
If \(y \notin A\text{,}\) then \(\tilde{f}_1(y)\) and \(\tilde{f}_2(y)\) lie in different sheets \(V_{\alpha_1} \ne V_{\alpha_2}\text{,}\) so \(\tilde{f}_1\) and \(\tilde{f}_2\) send a neighbourhood of \(y\) into disjoint sheets, meaning they disagree on that neighbourhood. Hence \(Y \setminus A\) is open.
Lemma 2.9. Homotopy lifting.
Let \(p \colon \tilde{X} \to X\) be a covering map, \(H \colon Y \times I \to X\) a homotopy, and write \(f = H(-,0)\text{.}\) Given a lift \(\tilde{f} \colon Y \to \tilde{X}\) of \(f\text{,}\) there is a unique map \(\tilde{H} \colon Y \times I \to \tilde{X}\) lifting \(H\) with \(\tilde{H}(y,0) = \tilde{f}(y)\text{.}\)
Proof.
See Hatcher, Algebraic Topology, p. 30. Uniqueness follows from Lemma 2.8.
Corollary 2.10. Path and homotopy lifting.
Let \(p \colon \tilde{X} \to X\) be a covering map.
-
Path lifting. Every path \(\alpha \colon I \to X\) starting at \(x_0\) lifts uniquely to a path \(\tilde{\alpha} \colon I \to \tilde{X}\) starting at any chosen \(\tilde{x}_0 \in p^{-1}(x_0)\text{.}\)
-
Homotopy lifting. Every path-homotopy \(H \colon I \times I \to X\) between paths starting at \(x_0\) lifts uniquely to a path-homotopy \(\tilde{H} \colon I \times I \to \tilde{X}\text{,}\) given a starting lift \(\tilde{x}_0\text{.}\)
Exercise 2.11.
Derive Corollary 2.10 from Lemma 2.9.
Proposition 2.12.
Let \(p \colon \tilde{X} \to X\) be a covering map with \(p(\tilde{x}_0) = x_0\text{.}\) The induced homomorphism \(p_* \colon \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0)\) is injective.
Proof.
Suppose \([\tilde{f}]\) is in the kernel, so \(p \circ \tilde{f} \simeq_p c_{x_0}\) via some \(H\text{.}\) By Corollary 2.10 we can lift \(H\) to \(\tilde{H}\) starting at \(\tilde{x}_0\text{.}\) The lift \(\tilde{H}\) is a path-homotopy from \(\tilde{f}\) to the constant loop at \(\tilde{x}_0\text{,}\) so \([\tilde{f}] = 0\text{.}\)
We will see a refined version of Proposition 2.12 in Chapter 3.
Definition 2.13.
Let \(p \colon \tilde{X} \to X\) be a covering map with \(p(\tilde{x}_0) = x_0\text{.}\) Then there is a map
\begin{equation*}
\phi \colon \pi_1(X, x_0) \to p^{-1}(x_0), \quad [f] \mapsto \tilde{f}(1)
\end{equation*}
which sends a loop class to the endpoint of its lift starting at \(\tilde{x}_0\text{.}\)
The map \(\phi\) is well-defined by Corollary 2.10.
Proposition 2.14.
Let \(p \colon \tilde{X} \to X\) be a covering map.
-
If \(\tilde{X}\) is path-connected, then \(\phi\) is surjective.
-
If \(\tilde{X}\) is simply connected, then \(\phi\) is bijective.
Proof.
Surjectivity. Given \(\tilde{x}_1 \in p^{-1}(x_0)\text{,}\) choose a path \(\tilde{\alpha}\) in \(\tilde{X}\) from \(\tilde{x}_0\) to \(\tilde{x}_1\text{.}\) Then \(f = p \circ \tilde{\alpha}\) is a loop in \(X\) at \(x_0\) with \(\phi([f]) = \tilde{x}_1\text{.}\)
Injectivity when \(\tilde{X}\) is simply connected. Suppose \(\phi([f]) = \phi([g])\text{,}\) i.e. the lifts \(\tilde{f}\) and \(\tilde{g}\) (both starting at \(\tilde{x}_0\)) have the same endpoint. Since \(\tilde{X}\) is simply connected, there is a path-homotopy \(\tilde{H}\) between \(\tilde{f}\) and \(\tilde{g}\text{.}\) Then \(p \circ \tilde{H}\) is a path-homotopy between \(f\) and \(g\text{,}\) so \([f] = [g]\text{.}\)
