In complex K theory, for any compact space X, Chern character $Ch$ induces a ring isomorphism $K_\mathbb{C}(X) \otimes \mathbb{Q} \rightarrow H^{even}(X;\mathbb{Q})$. For $X=S^2$, $K_\mathbb{C}(S^2) \otimes \mathbb{Q} \simeq \mathbb{Q} \oplus \mathbb{Q}$. Hence in this case, $Ch_0(S^2)$ and $Ch_1(S^2)$ correspond to these two $\mathbb{Q}$ respectively.
Now consider the real K theory, $K_\mathbb{R}(S^1) \simeq \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. Here is my question, is there a Cech(Singular) Cohomology invariant corresponds to the second factor $\mathbb{Z}/2\mathbb{Z}$? Is it a characteristic class?
Let $E \to S^1$ be a real vector bundle of rank $r+1$. It follows from obstruction theory that $E \cong L\oplus\varepsilon^r$ where $L$ is a real line bundle. As $H^1(S^1; \mathbb{Z}_2) \cong \mathbb{Z}_2$, there are two real line bundles on $S^1$ up to isomorphism, the trivial one $\varepsilon^1$ which has $w_1(\varepsilon^1) = 0$, and a non-trivial one $\gamma$ (sometimes called the Möbius bundle) which has $w_1(\gamma) \neq 0$.
Every element of $KO(S^1)$ can be written as $[E] - s$ for some vector bundle $E \to S^1$ and $s \geq 0$; here $s = [\varepsilon^s]$. By the above, $[E] - s = [L\oplus\varepsilon^r] - s = [L] + [\varepsilon^r] - s = [L] + r - s$. You can check that the map $\phi : KO(S^1) \to \mathbb{Z}\oplus\mathbb{Z}_2$ given by $\phi([E] - s) = (r - s, w_1(L))$ is an isomorphism of rings.