I think your last sentence/question is not well posed. It is well known that there are flat manifolds admitting plenty of non-flat vector bundles. E.g. see J. Smillie, "Flat manifolds with non-zero euler characteristics".

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I think your last sentence/question is not well posed. It is well known that there are flat manifolds admitting plenty of non-flat vector bundles. E.g. see J. Smillie, "Flat manifolds with non-zero Euler characteristics".

Sorry about the commentlike answer but I do not have enough points to properly comment.