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Question whether 2 spaces are diffeomorphic

  1. Nov 3, 2009 #1
    Suppose I have two embeddings of the circle into the 3 sphere. Is S3-minus the first image diffeomorphic to S3 - the image of the second?
     
  2. jcsd
  3. Nov 4, 2009 #2

    Office_Shredder

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    The answer is no. This is actually a knot theory concept.... an embedding of S1 in R3 is a knot, which are all homeomorphic to each other (generally taken that a knot is a homeomorphic image of S1 in R3. But in R3, there may not be a homeomorphism of R3 to itself that maps one to the other. There are some classic examples of knots that are not equivalent (knots tend to be embeddings) but I don't remember any off the top of my head, I'm sure a google search will reveal them
     
  4. Nov 5, 2009 #3
    Look at an unknot and a trefoil knot and compare the fundamental group of the complement.
     
  5. Nov 5, 2009 #4
    Thanks I will do tyhat.

    It is interesting because the first homology is just Z - by Alexander duality or a Meyer-Vietoris sequence argument.
     
    Last edited: Nov 5, 2009
  6. Nov 5, 2009 #5
    If I'm getting this right, the fundamental group of the complement of a non-trivial knot should be a bouquet of circles-type situation; i.e. you have some generators with some anti-commutation relations, and the commutator kills all of it.
     
  7. Nov 6, 2009 #6
    That has to right. And there is no torsion in the group mod its commutator subgroup.
    Do you know the generqator and relations for one of these?
     
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