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.
For the complex spinor bundle $\textbf{S} \to M^{2n}$ you can prove using the splitting principle, that $$ ch(\textbf{S}) = \prod\limits_{i=1}^{n} e^{x_i/2} + e^{-x_i/2}$$ where the $x_i$ are Chern roots for $TM \otimes \mathbb{C}$. See e.g. in "The Atiyah-Singer Index Theorem" by P. Shanahan page 55 and following. For $n=2$ this can be spelled out (in degree 2 and 4) \begin{align*} c_1(\textbf{S}) &= 0\\ \frac{c_1(\textbf{S})^2}{2}-c_2(\textbf{S}) &= \frac{1}{2}p_1(TM) \end{align*} So you have $c_2(\textbf{S}) = -\frac{1}{2}p_1(TM)$. After integration this becomes an integer. For the sphere it is zero and for the $K3$ surface it should be $-24$. Using that $16 = \mathrm{sign}(K3) = \frac{1}{3}\int_{K3} p_1(TK3)$.