[proofplan] For a group homomorphism between groups, injectivity is equivalent to triviality of the kernel, so it suffices to prove $\ker(p_*) = \{e\}$. A class $[\tilde{\gamma}] \in \ker(p_*)$ is represented by a loop $\tilde{\gamma}$ in $\tilde{X}$ based at $\tilde{x}_0$ whose image $\gamma = p \circ \tilde{\gamma}$ is null-homotopic in $X$ rel basepoint. A null-homotopy of $\gamma$ is a path homotopy from $\gamma$ to the constant loop $c_{x_0}$. Applying the [Lifted Path Homotopy theorem](/theorems/1888) to this homotopy — with the chosen lifts of $\gamma$ and $c_{x_0}$ being $\tilde{\gamma}$ itself and the constant loop $c_{\tilde{x}_0}$ — lifts the null-homotopy to a path homotopy in $\tilde{X}$ from $\tilde{\gamma}$ to $c_{\tilde{x}_0}$. Hence $[\tilde{\gamma}] = [c_{\tilde{x}_0}] = e$ in $\pi_1(\tilde{X}, \tilde{x}_0)$. [/proofplan] [step:Reduce injectivity to triviality of the kernel] The induced map $p_*: \pi_1(\tilde{X}, \tilde{x}_0) \to \pi_1(X, x_0)$ is a group homomorphism (by general functoriality of $\pi_1$ applied to the continuous map $p: (\tilde{X}, \tilde{x}_0) \to (X, x_0)$). For any group homomorphism $\varphi: G \to H$, $\varphi$ is injective if and only if $\ker(\varphi) = \{e_G\}$. It therefore suffices to prove \begin{align*} \ker(p_*) = \{[c_{\tilde{x}_0}]\}. \end{align*} [/step] [step:Take an arbitrary kernel element and unpack what it means] Let $[\tilde{\gamma}] \in \ker(p_*)$, represented by a continuous loop \begin{align*} \tilde{\gamma}: I \to \tilde{X}, \quad \tilde{\gamma}(0) = \tilde{\gamma}(1) = \tilde{x}_0. \end{align*} Set $\gamma := p \circ \tilde{\gamma}: I \to X$. Then $\gamma$ is a loop in $X$ based at $x_0 = p(\tilde{x}_0)$, and $[\gamma] = p_*([\tilde{\gamma}]) = [c_{x_0}]$ in $\pi_1(X, x_0)$. This means there exists a continuous path homotopy \begin{align*} H: I \times I &\to X \end{align*} with $H(-, 0) = \gamma$, $H(-, 1) = c_{x_0}$, $H(0, s) = x_0$ for all $s \in I$, and $H(1, s) = x_0$ for all $s \in I$. [guided] By definition, $[\gamma] = [c_{x_0}]$ in $\pi_1(X, x_0)$ means $\gamma$ and $c_{x_0}$ are path-homotopic in $X$ rel basepoint. Unpacking this, there is a continuous map $H$ from the square $I \times I$ to $X$ that interpolates between $\gamma$ and the constant loop $c_{x_0}$, with both vertical edges stuck at $x_0$ because the homotopy is rel basepoint (basepoint-fixing). Our goal is to upgrade this homotopy in $X$ to a homotopy in $\tilde{X}$ that witnesses $[\tilde{\gamma}] = [c_{\tilde{x}_0}]$. The tool for this is the Lifted Path Homotopy theorem, which says: a path homotopy downstairs lifts to a path homotopy upstairs, provided we have specified lifts of the two paths. We have $\tilde{\gamma}$ as a lift of $\gamma$; we need a lift of $c_{x_0}$ starting at $\tilde{x}_0$. [/guided] [/step] [step:Identify the required lift of the constant loop $c_{x_0}$] The constant loop $c_{\tilde{x}_0}: I \to \tilde{X}$, $t \mapsto \tilde{x}_0$, is continuous, satisfies $p \circ c_{\tilde{x}_0} = c_{x_0}$ (since $p(\tilde{x}_0) = x_0$), and starts at $\tilde{x}_0$. Hence $c_{\tilde{x}_0}$ is a continuous lift of $c_{x_0}$ starting at $\tilde{x}_0$. Similarly, $\tilde{\gamma}: I \to \tilde{X}$ is a continuous lift of $\gamma = p \circ \tilde{\gamma}$ and starts at $\tilde{x}_0$. [/step] [step:Apply the Lifted Path Homotopy theorem to obtain a path homotopy in $\tilde{X}$] We apply the [Lifted Path Homotopy theorem](/theorems/1888) to the data: - Covering map $p: \tilde{X} \to X$ (given). - Paths $\gamma$ and $c_{x_0}: I \to X$, which are homotopic as paths (rel endpoints) via $H$ constructed in Step 2 (both are loops at $x_0$, so the common endpoints are $x_0, x_0$). - Lifts $\tilde{\gamma}$ and $c_{\tilde{x}_0}$ of $\gamma$ and $c_{x_0}$ respectively, both starting at $\tilde{x}_0 \in p^{-1}(x_0)$ (verified in Step 3). The theorem concludes that $\tilde{\gamma}$ and $c_{\tilde{x}_0}$ are homotopic as paths in $\tilde{X}$ — that is, there exists a continuous path homotopy from $\tilde{\gamma}$ to $c_{\tilde{x}_0}$ rel endpoints $\tilde{x}_0, \tilde{x}_0$. Additionally, the "in particular" clause of the theorem gives $\tilde{\gamma}(1) = c_{\tilde{x}_0}(1) = \tilde{x}_0$, which is consistent with $\tilde{\gamma}$ being a loop at $\tilde{x}_0$. Therefore $[\tilde{\gamma}] = [c_{\tilde{x}_0}]$ in $\pi_1(\tilde{X}, \tilde{x}_0)$, i.e., $[\tilde{\gamma}]$ is the identity element of $\pi_1(\tilde{X}, \tilde{x}_0)$. [guided] The Lifted Path Homotopy theorem requires three inputs: a downstairs path homotopy $H$, and two lifts $\tilde{\gamma}, \tilde{\gamma}'$ of the endpoints of $H$ starting at the **same** point $\tilde{x}_0 \in p^{-1}(x_0)$. We have all three: $H$ was produced in Step 2, $\tilde{\gamma}$ is given as a loop based at $\tilde{x}_0$, and $c_{\tilde{x}_0}$ was constructed in Step 3 as a lift of $c_{x_0}$ starting at $\tilde{x}_0$. Why does the "both lifts start at the same point" hypothesis hold? This is the subtle check. The hypothesis $[\tilde{\gamma}] \in \pi_1(\tilde{X}, \tilde{x}_0)$ says $\tilde{\gamma}(0) = \tilde{x}_0$, and the constant loop $c_{\tilde{x}_0}$ trivially starts at $\tilde{x}_0$. Both start at $\tilde{x}_0$, as required. The output of the theorem is a path homotopy in $\tilde{X}$ from $\tilde{\gamma}$ to $c_{\tilde{x}_0}$ rel the common endpoints. In the language of the fundamental group, this is exactly the statement $[\tilde{\gamma}] = [c_{\tilde{x}_0}] = e$ in $\pi_1(\tilde{X}, \tilde{x}_0)$. It is worth noting where covering-space structure is used: the theorem we just applied is itself a consequence of the [Homotopy Lifting Lemma](/theorems/1887). Injectivity of $p_*$ is therefore not a formal group-theoretic accident — it genuinely uses the local homeomorphism structure of $p$ to lift homotopies. [/guided] [/step] [step:Conclude that $p_*$ is injective] We have shown that every $[\tilde{\gamma}] \in \ker(p_*)$ equals the identity element $[c_{\tilde{x}_0}]$ of $\pi_1(\tilde{X}, \tilde{x}_0)$. Hence $\ker(p_*) = \{e\}$, and by Step 1 this is equivalent to injectivity of $p_*$. This completes the proof. [/step]