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Given a spin structure on a oriented Riemannian manifold $(M,g)$, a spinor is a section of the spinor bundle $\pi:\mathbf{S}\to M$. I am trying to calculate the characteristic classes of the spinor bundle, in particular when $M$ is a 4-manifold.

In this case, the Dold-Whitney theorem says that bundles over $M^4$ are classified topologically by the second Stiefel-Whitney class and the first Pontryagin class. Note that the space of metrics on $M$ is convex (and hence contractible), so all spinor bundles on $M$ are isomorphic.

I am particularly interested in the cases of $S^4$ and a K3 surface. $S^4$ has no second cohomology, so the second Stiefel-Whitney class is trivial. The first Pontryagin class $p_1(\mathbf{S})\in H^4(S^4;\mathbb{Z})=\mathbb{Z}$ will correspond to some integer, but I'm not sure which one.

Thanks for your help.

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