I'm trying to understand the relationship between different generators of the $K$-theory group of $S^2$. Part of my curiosity comes from reading this discussion about characteristic classes.
The $K$-theory group $K^0(S^2)$ is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$. We may choose the first copy of $\mathbb{Z}$ to be generated by the trivial bundle $\varepsilon_{\mathbb{C}}^1=\mathbb{C}\times S^2$. For the second generator, there are two choices that I have seen in the literature, and I would like to understand how they relate to each other:
- The tautological bundle $H$ over $\mathbb{C}P^1\cong S^2$;
- The positive and negative spinor bundles $\mathcal{S}^+$ and $\mathcal{S}^-$ over $S^2$.
The first one is discussed in some detail here on nLab. The second one is used for example in section 2.16 of this paper by Baum and van Erp.
Let us fix a diffeomorphism $\psi\colon S^2\to\mathbb{C}P^1$. Then both $H$ and $\mathcal{S}^{\pm}$ can be viewed as line bundles over $S^2$, all with non-trivial first Chern classes I believe.
Question 1: Is $H$ isomorphic either to $\mathcal{S}^+$ or $\mathcal{S}^-$? Is there a canonical way to identify them, starting from a choice of $\psi$?
Looking at the answers to this question, it seems that isomorphism classes of line bundles correspond precisely to their first Chern classes. So perhaps this boils down to:
Question 2: Is $c_1(H)$ always equal to either $c_1(\mathcal{S}^\pm)$? How does this depend on $\psi$?
Comment: I think perhaps something similar can be done for $S^{2n}$, or maybe even $S^n$ for all $n$, but for now I'm just trying to understand $S^2$. I wonder if there is a reference discussing all this.