the lifting criterion
Suppose you have a covering p: X~ -> X stacked over a base, and a map f: Y -> X from some other space coming up from below. The natural question: can you raise f to a map f~: Y -> X~ landing in the cover, with p ∘ f~ = f? Sometimes yes, sometimes no, and the obstruction is purely group-theoretic. The lifting criterion is the exact statement of when the lift exists, and it is the bridge that turns covering spaces into a dictionary about subgroups of pi_1.
Let p: (X~, x~_0) -> (X, x_0) be a covering with X~ path-connected, and let f: (Y, y_0) -> (X, x_0) be a based map with Y path-connected and locally path-connected. Then a based lift f~: (Y, y_0) -> (X~, x~_0) with p ∘ f~ = f exists if and only if the image of f on fundamental groups sits inside the image of the cover: f_*(pi_1(Y, y_0)) is contained in p_*(pi_1(X~, x~_0)) as subgroups of pi_1(X, x_0). When the lift exists it is unique (given the basepoint choice), because two lifts agreeing at one point agree everywhere by the uniqueness of path lifting. The proof is constructive: to compute f~(y), pick a path from y_0 to y, push it down by f, lift that path in the cover starting at x~_0, and read off the endpoint — the subgroup condition is exactly what makes this independent of the chosen path.
This single criterion organizes the whole theory. Taking Y to be another cover gives the comparison of covers and the Galois correspondence; taking Y = X~ and f = p recovers deck transformations; taking Y simply connected (pi_1 trivial) shows every map from a simply connected space lifts, which is why the universal cover receives a lift of any map. It also instantly re-proves pi_1(S^1) = Z's injectivity and the unique-lifting facts behind winding numbers. Caveat worth stating: local path-connectedness of Y is a genuine hypothesis, not decoration — drop it and the constructed lift can fail to be continuous.
Try to lift the identity map S^1 -> S^1 through the double cover p: S^1 -> S^1, p(z) = z^2. Here f_*(pi_1) = Z but p_*(pi_1) = 2Z, and Z is not contained in 2Z, so no continuous lift exists — there is no continuous square-root function on the whole circle, which the criterion predicts at a glance.
Z not in 2Z forecasts the failure of a global square root — covering-space obstruction made arithmetic.
The criterion is about BASED maps; an unbased lift may exist when a based one does not, after conjugating the subgroup by choosing a different basepoint in the fiber. Keep track of basepoints, because the subgroup p_*(pi_1) is only defined up to conjugacy without one.