Let \(\tilde{X}\) and \(\tilde{Y}\) be simply-connected covering spaces of the path-connected, locally path-connected spaces \(X\) and \(Y\). Show that if \(X \simeq Y\) then \(\tilde{X} \simeq \tilde{Y}\).
Solution
Let \[ p_X:\widetilde X\to X, \qquad p_Y:\widetilde Y\to Y \] be the given simply connected covering spaces. Choose homotopy inverse maps \[ f:X\to Y, \qquad g:Y\to X \] with \[ g f\simeq\operatorname{id}_X, \qquad f g\simeq\operatorname{id}_Y. \]
1. The composite\[ f\circ p_X:\widetilde X\to Y \]lifts to a map\[ \widetilde f:\widetilde X\to\widetilde Y. \]
Since \(\widetilde X\) is simply connected, \[ (f p_X)_*\pi_1(\widetilde X)=0. \] This subgroup is contained in \[ (p_Y)_*\pi_1(\widetilde Y)=0. \] The covering-space lifting criterion therefore gives a lift after choosing compatible basepoints.
2. Similarly, \(g\circ p_Y\) lifts to a map\[ \widetilde g:\widetilde Y\to\widetilde X. \]
The same lifting criterion applies because \(\widetilde Y\) is simply connected.
3. The composite \(\widetilde g\widetilde f\) is homotopic to \(\operatorname{id}_{\widetilde X}\).
Let \[ H:X\times I\to X \] be a homotopy from \(g f\) to \(\operatorname{id}_X\). Then \[ H\circ(p_X\times\operatorname{id}_I):\widetilde X\times I\to X \] is a homotopy from \[ g f p_X=p_X\widetilde g\widetilde f \] to \(p_X\).
By the homotopy lifting property of the covering \(p_X\), lift this homotopy starting at the map \(\widetilde g\widetilde f\). Its terminal map \(u:\widetilde X\to\widetilde X\) satisfies \[ p_Xu=p_X. \] Choose the lifts \(\widetilde f,\widetilde g\) and the lifted homotopy so that one chosen basepoint is fixed at the terminal time. Then \(u\) and \(\operatorname{id}_{\widetilde X}\) are two lifts of \(p_X\) agreeing at that basepoint. Uniqueness of lifts gives \[ u=\operatorname{id}_{\widetilde X}. \] Hence \[ \widetilde g\widetilde f\simeq\operatorname{id}_{\widetilde X}. \]
4. Likewise,\[ \widetilde f\widetilde g\simeq\operatorname{id}_{\widetilde Y}. \]
Apply the same argument to a homotopy \[ f g\simeq\operatorname{id}_Y \] and the covering \(p_Y\).
5. Therefore\[ \boxed{\widetilde X\simeq\widetilde Y.} \]
The maps \(\widetilde f\) and \(\widetilde g\) are homotopy inverses by <1>3–<1>4.