All Questions

It's well known that for a complex vector bundle $E$, we have $$c_n(E)=e_n(E_\mathbb{R}) $$ But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class ...

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 ...