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What's the fundamental group of a punctured torus?

  1. Aug 24, 2010 #1
    The fundamental group of a torus is Z[tex]*[/tex]Z,then the fundamental group of a punctured torus is Z[tex]*[/tex]Z[tex]*[/tex]Z.

    But I've ever done a problem,it said a punctured torus can be continuously deformed into two cylinders glued to a square patch.Really?

    If that is right,then the fundamental group of punctured torus is Z[tex]*[/tex]Z.

    Which is right?Need help:smile:
     

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  3. Aug 24, 2010 #2

    quasar987

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    Yes, it's true... imagine making the hole bigger and bigger.. make it as big as you can withouth changing the topology. You're left with the two strips glued on a square patch.
     
  4. Aug 24, 2010 #3
    oh,i see.the fundamental group should be Z*Z. i consider an extra loop,which is the edge circle of the punctured hole. but now I know it's the 2 power of a generator.
     
  5. Aug 24, 2010 #4

    quasar987

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    The fundamental group of the torus is not Z*Z though, it is ZxZ.
     
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