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Thread: Trivial vector bundle problem

  1. #1
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    Trivial vector bundle problem

    I have a problem showing that any vector bundle $\displaystyle E \to [0,1]$ is trivial.

    The hint given with the exercise is considering the biggest number $\displaystyle t \in [0,1]$ so that $\displaystyle \pi^{-1}([0,t]) \to [0,t]$ is trivial, but i don't get how to do that, in fact I'm not even sure what that last statement means.

    So far I think I've shown that for some $\displaystyle t \in (0,1)$, we have a homeomorphism $\displaystyle \pi^{-1}([0,t]) \to [0,t]$, since there must exist a homeomorphism to an open neighbourhood of 0, and we can restrict that. Also, $\displaystyle [0,t]$ is homeomorphic to $\displaystyle [0,1]$. But where do I go from there?
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  2. #2
    Senior Member Tinyboss's Avatar
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    I think it's going to go something like

    1) Show such a t exists.

    2) Show t=1 (assume <1 and get a contradiction).

    I suspect that connectedness and/or compactness will come into play, but I can't see the answer immediately.
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