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Square lemma for Paths, Homotopy

  1. Feb 19, 2013 #1
    1. The problem statement, all variables and given/known data
    In Lee's "Topological Manifolds", there is a result on page 193 called "The Square Lemma" which states that if [itex] I[/itex] denotes the unit interval in [itex] \mathbb{R}, [/itex] [itex] X [/itex] is a topological space, [itex]F\colon I\times I\to X [/itex] is continuous, and [itex] f,g,h,k [/itex] are paths defined by
    [itex] f(s)=F(s,0),\ g(s)=F(1,s),\ h(s)=F(0,s),\ k(s)=F(s,1), [/itex]
    then [itex] f\cdot g\sim h\cdot k [/itex] where [itex] \cdot [/itex] denotes path concatonation and [itex] \sim [/itex] denotes homotopy rel [itex] \{0,1\}. [/itex]

    It seems to me that this is not guaranteed to be true if [itex] X [/itex] is not simply connected. Indeed if we take [itex] X=I\times I [/itex] and take [itex] F=\text{id}_{I\times I} [/itex], then I don't think that we have [itex] f\cdot g\sim h\cdot k [/itex].

    Am I wrong about this?
     
  2. jcsd
  3. Feb 19, 2013 #2

    Dick

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    What's not simply connected about I x I? And why don't you think fg and hk are homotopic? You can deform them both to a path across the diagonal of the square.
     
  4. Feb 19, 2013 #3
    Ah. Brain malfunction on my part. For some reason I was thinking of [itex] I\times I [/itex] as the boundary of the square and not the whole square. That's sorted then. Thanks!
     
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