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For my differential topology course I need to prove that every vector bundle over $S^1$ is either trivial or equivelent to the Moebius bundle.

I found this nice, elementary proof: Line bundles of the circle, but it uses that every vector bundle over $\mathbb{R}$ is trivial. The proofs of this statement I found all use parallel transport, which we haven´t defined in the lecture and which we therefore shouldn´t use. Does anybody have a proof not using parallel transport or any other sophisticated tool like cohomology theory?

suchter66
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    It's well known that fiber bundles over contractible spaces are trivial. Here's a link for a proof for bundles over Euclidean spaces: https://math.stackexchange.com/a/1085448/55622 – Oliver Jones Nov 25 '19 at 08:25
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    For vector bundles, you may need to add paracompactness. – Oliver Jones Nov 25 '19 at 08:33

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