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Inverse Function Thm. and Covering Maps.

  1. Jan 4, 2014 #1

    WWGD

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    Hi, All:

    Let ## f: X → Y ## be a differentiable map , so that ## Df(x)≠0 ## for all ##x## in ##X##. Then the inverse function
    theorem guarantees that every point has a neighborhood where ##f ## restricts to a homeomorphism.

    Does anyone know the conditions under which conditions a map like above is a covering map? I'm thinking of the case of the complex exponential ## e^z ## , with ##d/dz(e^z)=e^z ≠0## which is a covering map ## \mathbb C^2 → (\mathbb C-{0} ) ## , but I can't tell if the condition ## df(x)≠ 0 ## is enough to guarantee that ##f ## is a covering map, nor what conditions would make ##f ## into a covering map.

    Thanks for any Ideas.
     
  2. jcsd
  3. Jan 5, 2014 #2

    mathwonk

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  4. Jan 7, 2014 #3

    WWGD

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    Thanks, Mathwonk.
     
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