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Definition of Null Homotopic

  1. Apr 18, 2009 #1
    I have a question about the definition of null homotopic. My textbook (Functions of one Complex Variable I by John B. Conway) defines it as follows: If [tex]\gamma[/tex] is a closed rectifiable curve in a region G, then [tex]\gamma[/tex] is homotopic to zero if [tex]\gamma[/tex] is homotopic to a constant curve. My question is, if G is simply connected, then is [tex]\gamma[/tex] homotopic to ANY constant curve in G? This seems obvious to me, but I'm not sure if I should prove it or just state it.
     
    Last edited: Apr 18, 2009
  2. jcsd
  3. Apr 19, 2009 #2
    Here's my joke: When is a curve that is homotopic to zero not homotopic to zero?

    Answer: Let Dn be the open disk of radius 1 centered at n, and let G = union of D0 and D3. Then G is simply connected but not connected.

    Let gamma be a curve in D3. Then gamma is homotopic to zero because it is homotopic to the constant curve alpha(t)=3.

    However, gamma is not homotopic to the constant curve beta(t)=0, so one might say that gamma is not homotopic to "zero."
     
  4. Apr 23, 2009 #3
    Your G would not be simply connected :/ It has to be path connected and therefore connected to be simply connected.
     
  5. Apr 23, 2009 #4

    Hurkyl

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    If G~c and G~c', then you would need to have c~c'....
     
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