Any continuous map $f: X_1 \to X_2$ "lifts" to a map $\tilde f: \tilde X_1 \to \tilde X_2$ (provided that $X_1$ and $X_2$ have universal covers).
The space $\tilde X_1$ is certainly path-connected and locally path-connected and we have $$(f\circ p_1)_*(\pi_1(\tilde X_1))=\{0\}=(p_2)_*(\pi_1(\tilde X_2)).$$ Then the lifting criterion ensures the existence of a lift $\ \tilde f: \tilde X_1 \to \tilde X_2$. Is this a very basic fact I somehow forgot, or have I made some glaring error?