Relation between a map and its lifting

1. Sep 20, 2011

yaa09d

I have the following question:
Let $\mathbb{D}$ denote the unit disk.
Let $f:X_1 \longrightarrow X_2$ be a continuous mapping between Riemann Surfaces.
Let $\pi_1 : \mathbb{D} \longrightarrow X_1$ , and $\pi_2 : \mathbb{D} \longrightarrow X_2$ be the universal covering spaces of $X_1$ and $X_2$, respectively. A lifting of $f$ is a continuous map $\tilde{f}: \mathbb{D}\longrightarrow \mathbb{D}$ such that $f\circ \pi_1=\pi_2\circ \tilde{f}.$

The question is to show if $f$ is homeomorphism, then so is $\tilde{f},$ or to give a counterexample.

Any help will be appreciated.

Thank you.