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Contractible curves

  1. Dec 27, 2013 #1
    I read some text to find it's definition
    Is it possible to tell me it's definition?


    I read below statements about local and global geometry and I didn't understand it. is it possible tell me it.
    "If M ( a manifold) has a trivial topology, a single neighborhood can be extended globally, and geometry is indeed trivial; but if M contains non-contractible curves, such as extension may not be possible."
     
  2. jcsd
  3. Dec 27, 2013 #2

    tiny-tim

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    hi sadegh4137! :smile:

    (if C1 is the unit circle, ie [0,1] with 0 and 1 the same point)

    a closed curve f:C1 -> M on a manifold M is contractible if it can be contracted to a point,

    ie if there's a continuous function g:[0,1] -> C1M such that each g(t) is continuous, g(0) is a single point, and g(1) is f

    eg the surface of a torus is not contractible, since a circle that "loops" the hole cannot be shrunk to a point!
     
  4. Jan 4, 2014 #3

    WWGD

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    Sorry to nitpick, Tiny Tim, but I think it is important to note that the contraction must be done

    within the space ( I thinks this follows from your definition of g , but I think it is important to say it any way, since I think it brings room for confusion ), in case the space is embedded somewhere else. As an example, if we have

    ## S^1 ## embedded in ## \mathbb R^2 ## , then note that ## S^1 ## --and any curve in it--

    can be contracted to a point if we can work in ## \mathbb R^2 ## , but not so if, while doing the deformation , we must stay within ## S^1 ## .
     
    Last edited: Jan 5, 2014
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