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Extending Trivializations and Structure Groups

  1. Jul 23, 2010 #1
    Hi, Everyone:

    LetB: E-->X be a line bundle, with scructure group G and X has a CW -decomposition.

    I am trying to understand why/how, if the structure group G of B is connected,
    then any trivialization over the 0-skeleton of X can be extended to a trivialization
    of the 1-skeleton.

    I understand that for every k-cell f:D^k --.X (D^k is the k-disk) , the
    pullback bundle is trivial (by contractibility of D^k), but I don't see how/why
    the connectedness of G alllows us to extend a given trivialization from the
    0-skeleton to the 1-skeleton.

    There is also a mention of a canonical trivialization over the cells. Anyone
    know what that is.?

    Thanks.
     
  2. jcsd
  3. Jul 23, 2010 #2
    I wonder if someone knows examples of trivializations (i.e., global sections)
    that do not extend beyond the 0-skeleton, maybe the 1-skeleton. It seems to
    come down to extending maps from the interior of a cell to its boundary, maybe
    with retractions.
    Am I on the right track.?
     
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