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As a newbie to vector bundles, it seems like all vector bundles I have run into ( not that many, I admit) need only two charts to be trivialized; one of these charts will contain the "trouble point" (the point that prevents the top space from being a global product), and one chart that will not contain the trouble point ( I am thinking of the Möbius band over the circle here ).

It also seems like the trivializations ultimately consist of unwrapping the twist in the trouble point in the top( again, in here, in the case of the Möbius bundle, I am thinking of the lift of (1,0) in S^1 ) , unwrapping , and flattenning.

Is this an at least informally correct generalization of vector bundles ( or, at least of plane bundles, i.e., 1-bundles? )

Thanks.

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    Perhaps I am alone, but I do not understand this question. What are you trying to generalize vector bundles to?2011-01-30
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    I think "generalization of vector bundles" is not really the right phrase to use here. There are certainly examples of vector bundles in which more than two charts are required. The question might be better phrased as a request for such examples.2011-01-30
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    @Dan: tangentially, I wonder if the invariant of a vector bundle giving the minimal number of charts one needs to trivialize it has been studied---something like LS category...2011-02-04

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