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Math Help - Retract (algebraic topology)

  1. #1
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    Retract (algebraic topology)

    Hi everyone, excuse me about my bad english but it isn't my natural language.
    I have tried so much but I canīt solve the next problem:


    Let  K be a closed orientable surface. Let  \alpha \subset K a loop such that K - \alpha is connected. Show that \alpha is retract of K.


    I would like to know how to solve this.I hope sombody could help me. Thanks anyway.
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  2. #2
    Super Member Rebesques's Avatar
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    Choose two distinct points x,y\in (K-a)^o. The homogeneity lemma (cf Milnor's "Topology from the Differentiable Viewpoint") guarantees that there exists a
    diffeomorphism of K-a onto itself that maps one point to the other, and that this diffeomorphism is smoothly isotopic to the identity.
    Last edited by Rebesques; December 29th 2010 at 03:04 PM.
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  3. #3
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    cut off one of the two generator circles of a torus, the resulting space is a cylinder which is connected, while a torus does not retract to a circle since its fundamental group is Z x Z. Did I understand your question correctly?
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  4. #4
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    Quote Originally Posted by xxp9 View Post
    cut off one of the two generator circles of a torus, the resulting space is a cylinder which is connected, while a torus does not retract to a circle since its fundamental group is Z x Z. Did I understand your question correctly?
    A "deformation retract" and a "retract" are not the same thing. Your argument works if the question asked about a "deformation retract".

    In other words, a torus does not deformation retract to a circle, but it retracts to a circle.
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