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Math Help - Simple connectivity and finite coverings

  1. #1
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    Simple connectivity and finite coverings

    Hey everyone, this is my first post so please say something if it's in the wrong place or if I commit some faux pau :-)

    So, the basic question is "Why does simple connectivity (in a topological sense; the fundamental group of some manifold M is trivial) imply there cannot be a finite covering of M?" The actual manifold we (my advisor and I) were discussing was CP^2, and she was trying to explain to me why were needed a branched covering. I'll deal with that later but right now I just need to understand this.

    I guess I don't understand because I can think of things I would think are counter-examples: The closed disk should probably have a finite covering and it is simply connected. So any enlightenment would be great; I am new to topology but I really need to understand this stuff.

    Thanks very much!
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by cduston View Post
    Hey everyone, this is my first post so please say something if it's in the wrong place or if I commit some faux pas \leftarrow \color{red}\bold{\text{here is one}} :-)

    So, the basic question is "Why does simple connectivity (in a topological sense; the fundamental group of some manifold M is trivial) imply there cannot be a finite covering of M?" The actual manifold we (my advisor and I) were discussing was CP^2, and she was trying to explain to me why were needed a branched covering. I'll deal with that later but right now I just need to understand this.

    I guess I don't understand because I can think of things I would think are counter-examples: The closed disk should probably have a finite covering and it is simply connected. So any enlightenment would be great; I am new to topology but I really need to understand this stuff.

    Thanks very much!
    ..
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  3. #3
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    If it were true that simple connectivity implies no finite covering, this would mean that compact spaces could not be simply connected Ė which is clearly absurd, because there certainly are compact spaces which are simply connected. Maybe this is true only for the particular type of manifold youíre considering. It definitely isnít true for all topological spaces in general.
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  4. #4
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    Ok good!

    Ok so I'm not totally insane, my counterarguments make (at least some) sense. Does anyone know, then, why the complex projective plane CP_2 does not admit a finite covering? My professor (who is a fairly well-known mathematical physicist...I don't believe she should be wrong, although this was an e-mail correspondence so there could be some confusion there) said that a finite covering of CP_2 would correspond to a normal subgroup of the fundamental group, but since the fundamental group is trivial for CP_2 such a covering cannot exist. Any thoughts or directions to turn?

    Thanks again.

    EDIT: I should add that I am talking about covering SPACES. I'm not even sure if that is different then something like "a union of finite open sets" or something, but if there is I am trying to ask specifically about finite covering spaces.
    Last edited by cduston; July 3rd 2008 at 06:21 AM. Reason: clarification
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