I have the following question:(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Relation between a map and its lifting

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