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Relation between a map and its lifting

  1. Sep 20, 2011 #1
    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.
  2. jcsd
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