Show that if finite connected CW complexes \(X\) and \(Y\) are homotopy equivalent, then their universal covers \(\tilde{X}\) and \(\tilde{Y}\) are proper homotopy equivalent.
Solution
Let \[ f:X\to Y, \qquad g:Y\to X \] be homotopy inverse maps between finite connected CW complexes. Choose basepoints and lifts \[ \widetilde f:\widetilde X\to\widetilde Y, \qquad \widetilde g:\widetilde Y\to\widetilde X. \] Since \(f_*\) is an isomorphism on fundamental groups, after identifying the deck groups through \(f_*\) the lift \(\widetilde f\) is equivariant: \[ \widetilde f(\gamma x)=f_*(\gamma)\widetilde f(x). \] Similarly for \(\widetilde g\).
We first show these lifts are proper. Let \(K_X\subset\widetilde X\) and \(K_Y\subset\widetilde Y\) be finite unions of lifted closed cells mapping onto \(X\) and \(Y\). Every compact subset of a universal cover of a finite CW complex meets only finitely many deck translates of such a fundamental finite subcomplex. By equivariance, the inverse image under \(\widetilde f\) of a finite union of translates of \(K_Y\) lies in a finite union of translates of \(K_X\). Hence inverse images of compact sets are compact, so \(\widetilde f\) is proper; likewise \(\widetilde g\).
Choose homotopies \[ gf\simeq\operatorname{id}_X, \qquad fg\simeq\operatorname{id}_Y. \] They lift to homotopies \[ \widetilde g\widetilde f\simeq\operatorname{id}_{\widetilde X}, \qquad \widetilde f\widetilde g\simeq\operatorname{id}_{\widetilde Y} \] after the endpoint lifts are chosen compatibly. These lifted homotopies are proper: they are equivariant lifts of homotopies on the compact finite CW quotients \(X\times I\) and \(Y\times I\), so the same finite-fundamental-domain argument applies uniformly in the parameter.
Thus \(\widetilde f\) and \(\widetilde g\) are proper maps and proper homotopy inverses. Therefore \[ \boxed{\widetilde X\simeq_{\mathrm{proper}}\widetilde Y.} \]