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Computing $mathrm{Vect}_k(M)$?

Mathematics Asked by KoKo on December 1, 2020

$mathrm{Vect}_k(M)$ is the isomorphism classes of real $k$ rank vector bundles over $M$.
In Bott-Tu book they give only an example:
For contractible manifolds it is $mathrm{Vect}_k(M)$ is a point.
I don’t think I could understand well only by this example.
Could you give me other nontrivial examples?
In Exercise 6.10 of this book.
It is asked: Compute $mathrm{Vect}_k(S^1)$.

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