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I am confused about the existence of a local spinor bundle.

My question is that if a Riemannian manifold $M$ is not spin, why does there exist a local spinor bundle over all sufficiently small open subsets of $M$?

I'm not sure if it is too easy on Mathoverflow. Could you give me some help with the details? Thanks in advance.

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    $\begingroup$ I'm not sure I understand your question. Sufficiently small (say contractible ones), open subsets of $M$ are themselves spin manifolds. So doesn't your first sentence already answer your second sentence? $\endgroup$
    – Igor Khavkine
    Commented Apr 12, 2022 at 10:07
  • $\begingroup$ @IgorKhavkine I edited my question again. $\endgroup$
    – Radeha Longa
    Commented Apr 13, 2022 at 1:12
  • $\begingroup$ Are you talking about how you can construct the bundle directly out of inhomogeneous sums of differential forms as in J. Math. Phys. 37, 3882 (1996) aip.scitation.org/doi/10.1063/1.531607 ? $\endgroup$
    – Buzz
    Commented Apr 13, 2022 at 1:49
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    $\begingroup$ The existence of a spin-structure and thus an associated spinor bundle is purely topological and obstructed by the first and second Stiefel-Whitney-class, which vanish in your situation, since you have sufficiently small open subsets, which we can be choosen contractible. $\endgroup$
    – Julian Seipel
    Commented Apr 13, 2022 at 7:49

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