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Math Help - Homotopy equivalence and covering spaces

  1. #1
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    Homotopy equivalence and covering spaces

    Let Cov X and Cov Y be simply connected covering spaces of the path connected and locally path connected spaces X and Y respectively. Show that if X is homotopy equivalent to Y, then Cov X is homotopy equivalent to Cov Y.
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    MHF Contributor Drexel28's Avatar
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    Re: Homotopy equivalence and covering spaces

    Quote Originally Posted by kierkegaard View Post
    Let Cov X and Cov Y be simply connected covering spaces of the path connected and locally path connected spaces X and Y respectively. Show that if X is homotopy equivalent to Y, then Cov X is homotopy equivalent to Cov Y.
    This problem is extremely ugly, at least the way I did it when I did it out of Hatcher. Do you at least have any leads?
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    Re: Homotopy equivalence and covering spaces

    Not so far, except the functions that you have by definition of homotopy equivalence. For example, I don't know why we need that the spaces be locally path connected.
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    MHF Contributor Drexel28's Avatar
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    Re: Homotopy equivalence and covering spaces

    Quote Originally Posted by kierkegaard View Post
    Not so far, except the functions that you have by definition of homotopy equivalence. For example, I don't know why we need that the spaces be locally path connected.
    I can try to find the solution for you, but roughly you just write out what's going on, use the universal lifting property to get maps, \widetilde{X}\leftrightarrow\widetilde{Y}, show these are homotopic to deck transformations, and then use this to conclude.
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    Re: Homotopy equivalence and covering spaces

    Thanks, now I'm going to put together to solve the exercise. If I cannot find the solution, I'm going to reply.
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  6. #6
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    Re: Homotopy equivalence and covering spaces

    Ok, this is the approach so far. If p:\bar{X}\to X is a covering map, then p\times id_{I}:\bar{X}\times I\to X\times I is also a covering map. Moreover,since \bar{X} is simply connected, \bar{X}\times I is also simply connected and locally path-connected.Similarly with Y and \bar{Y} and I guess I can lift the homotopy but I'm not sure how to continue fron this point.

    Drexel28, can you help me?
    Last edited by kierkegaard; December 4th 2011 at 07:51 AM.
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