For cell complexes${}^1$ $X$ we have an isomorphism


$$
K^\*(X)\otimes \mathbb{Q}\cong H^{\*}(X;\mathbb{Q}),
$$


which is induced by the Chern character. 


What is the analogous statement for $KO(X)$?


${}^1$:Hatcher states finite, but I've seen arbitrary CW-complexes stated as well.


edit: The footnote seems wrong, as by the comments.



$K$ theory and singular cohomology
In the following, $X$ is a finite complex. The Adams operator $\psi^{-1}$ (complex conjugation) acts on $K^0(X)$. After inverting $2$, the group $KO^0(X) \otimes \Bbb Z[1/2]$ maps isomorphically to the subset of $K^0(X) \otimes \Bbb Z[1/2]$ fixed by $\psi^{-1}$.


We can then tensor this with $\Bbb Q$. Then $K^0(X) \otimes \Bbb Q \cong \prod\_{n \geq 0} H^{2n}(X;\Bbb Q)$. The operator $\psi^{-1}$ acts on this by fixing the factors with $n$ even and negating the factors with $n$ odd. The fixed set is $KO^0(X) \otimes \Bbb Q \cong \prod\_{m \geq 0} H^{4m}(X; \Bbb Q)$.


