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Null-homologous framing?

  1. Jan 21, 2014 #1

    WWGD

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    Hi, all:

    I'm trying to understand the meaning of the term "null-homotopic framing".

    Say K is a knot embedded in a manifold , and N(K) is a tubular neighborhood of K

    ( there is a theorem that a compact submanifold allows for the existence of a tubular

    neighborhood).

    I read about it here, under " motivation for the Legendrian conjecture " :

    http://electrichandleslide.wordpress.com/2013/05/17/the-legendrian-surgery-conjecture/

    A framing here is a choice ( up to isotopy, I believe; not specified in Rolfsen's book) of

    isomorphism between the tubular neighborhood N(K) and ## S^1 \times D^2 ##.

    (Though K can be any manifold for which a tubular neighborhood exists, e.g., maybe

    the normal bundle N(K) is trivial, or K is compact, embedded in some ambient manifold M)

    I understand the obvious meaning of null-homologous for a cycle in any space, meaning

    that the cycle is , e.g., a bounding cycle (cycle with non-trivial boundary) in simplicial homology,

    so homologically trivial, or different definitions for different choices of homology, but I cannot see what a

    null-homologous framing is. Any ideas?

    Thanks.
     
    Last edited: Jan 22, 2014
  2. jcsd
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