Extending Trivializations and Structure Groups

  • Thread starter Bacle
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
662
1
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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