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Fundamental group to second homology group

  1. May 2, 2010 #1


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    In a smooth compact 3 manifold there is an embedded loop - a diffeomorph of the circle

    Consider a torus that is the boundary of a tubular neighborhood of this loop.

    If the loop is not null homotopic does that imply that the torus is not null homologous?
  2. jcsd
  3. May 3, 2010 #2
    Why would it? Doesn't the torus clearly bound? (I.e. you've defined it as the boundary of an open set in your 3-manifold, right?)
  4. May 6, 2010 #3


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    yes. Stupid question.

    I am trying to understand how an element of the fundamental group can determine a homology class - but this element is null homologous though not null homotopic. The homology class would be 2 dimensional. For a moment I thought the torus might work - but that thought is as yopu pointed out - empty.

    Thanks for your reply though
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