Let Cov X and Cov Y be simply connected covering spaces of the path connected and locally path connected spaces X and Y respectively. Show that if X is homotopy equivalent to Y, then Cov X is homotopy equivalent to Cov Y.
Ok, this is the approach so far. If is a covering map, then is also a covering map. Moreover,since is simply connected, is also simply connected and locally path-connected.Similarly with and and I guess I can lift the homotopy but I'm not sure how to continue fron this point.
Drexel28, can you help me?