What is the meaning of $KO^{-1}(S^1)$?

Question

I am interested in the KO-theory of the circle $S^1$. In particular $KO^{-1}(S^1)$. Using the suspension theorem and reduced $K$-theory I can easily show that \begin{equation} KO^{-1}(S^1) \simeq KO^{-1}(pt) \oplus KO^{-2}(pt) \simeq \mathbb{Z}_2^2\,. \end{equation} My question is now what the interpretation is of these two $Z_2$s? I understand that they tell me that there are four different types of bundles, but is there a topological invariant that distinguishes them? For instance, can we use Stiefel-Whitney classes? Any reference where this is addressed would also be very welcome!

Answer 1

Note that $$KO^{-1}(S^1) \cong \widetilde{KO}^{-1}(S^1\sqcup\text{pt}) \cong \widetilde{KO}(\Sigma(S^1\sqcup\text{pt})) \cong \widetilde{KO}(S^2\vee S^1) \cong \widetilde{KO}(S^2)\oplus\widetilde{KO}(S^1) \cong \mathbb{Z}_2\oplus\mathbb{Z}_2.$$

The generator of $\widetilde{KO}(S^2)$ is the equivalence class of non-spin bundles which is represented by $\mathcal{O}(1) \to \mathbb{CP}^1$, viewed as a real rank two bundle over $S^2$.

The generator of $\widetilde{KO}(S^1)$ is the equivalence class of non-orientable bundles which is represented by $\gamma_1 \to \mathbb{RP}^1$, viewed as a real rank one bundle over $S^1$.