Let \(p: \tilde{X} \to X\) be the universal cover of a path-connected space \(X\). Show that under the isomorphism \(\pi_n(X) \approx \pi_n(\tilde{X})\), which holds for \(n \geq 2\), the action of \(\pi_1(X)\) on \(\pi_n(X)\) corresponds to the action of \(\pi_1(X)\) on \(\pi_n(\tilde{X})\) induced by the action of \(\pi_1(X)\) on \(\tilde{X}\) as deck transformations. More precisely, prove a formula like \(\gamma p_*(\alpha) = p_*\bigl(\beta_{\tilde{\gamma}}(\gamma_*(\alpha))\bigr)\) where \(\gamma \in \pi_1(X, x_0)\), \(\alpha \in \pi_n(\tilde{X}, \tilde{x}_0)\), and \(\gamma_*\) denotes the homomorphism induced by the action of \(\gamma\) on \(\tilde{X}\).
Solution
Let \(\widetilde x_0\in p^{-1}(x_0)\), and let \(\gamma\) be a loop at \(x_0\). Lift \(\gamma\) to a path \[ \widetilde\gamma:I\to\widetilde X, \qquad \widetilde\gamma(0)=\widetilde x_0. \] Its endpoint is \(T_\gamma(\widetilde x_0)\), where \(T_\gamma\) is the deck transformation corresponding to \(\gamma\).
Take \[ \alpha\in\pi_n(\widetilde X,\widetilde x_0),\qquad n\ge2. \] Applying \(T_\gamma\) gives \[ (T_\gamma)_*(\alpha)\in \pi_n(\widetilde X,T_\gamma\widetilde x_0). \] Change the basepoint back along \(\widetilde\gamma\). With the convention that \(\beta_{\widetilde\gamma}:\pi_n(\widetilde X,T_\gamma\widetilde x_0)\to \pi_n(\widetilde X,\widetilde x_0)\), the resulting class is \[ \beta_{\widetilde\gamma}(T_\gamma)_*(\alpha). \]
Projecting to \(X\), the path \(\widetilde\gamma\) projects to \(\gamma\), while \(pT_\gamma=p\). Hence the defining construction of the \(\pi_1(X)\)-action on \(\pi_n(X)\) gives the commutative formula \[ \boxed{ \gamma\cdot p_*(\alpha) =p_*\!\left(\beta_{\widetilde\gamma}(T_\gamma)_*(\alpha)\right). } \] If the opposite convention for change-of-basepoint maps is used, the same formula is written with \(\beta_{\widetilde\gamma}^{-1}\); the content is unchanged. Thus under \(p_*:\pi_n(\widetilde X)\cong\pi_n(X)\), the usual \(\pi_1(X)\)-action is precisely the action induced by deck transformations, with the necessary basepoint correction.