2 typo

For any (connective) spectrum $E$ one may rationalise it to get a rational spectrum $E_\mathbb{Q}$, and a map $E \to E_\mathbb{Q}$. Now rational spectra split as wedges of Eilenberg-Mac Lane spectra, so one may choose an isomorphism $E_\mathbb{Q}^*(X) \simeq H^*(X;\pi_{-*}(E)\otimes \mathbb{Q})$, and the rationalisation gives a map $$ch_E : E^*(X) \longrightarrow H^*(X;\pi_{-*}(E)\otimes \mathbb{Q}).$$

For complex K-theory this gives the Chern character, and for real K-theory it gives the Pontrjagin character.

Of course, is if $E$ is a ring spectrum so is $E_\mathbb{Q}$, and one must identify the induced ring structure on $H^*(X;\pi_{-*}(E)\otimes \mathbb{Q})$.

1

For any (connective) spectrum $E$ one may rationalise it to get a rational spectrum $E_\mathbb{Q}$, and a map $E \to E_\mathbb{Q}$. Now rational spectra split as wedges of Eilenberg-Mac Lane spectra, so one may choose an isomorphism $E_\mathbb{Q}^*(X) \simeq H^*(X;\pi_{-*}(E)\otimes \mathbb{Q})$, and the rationalisation gives a map $$ch_E : E^*(X) \longrightarrow H^*(X;\pi_{-*}(E)\otimes \mathbb{Q}).$$

For complex K-theory this gives the Chern character, and for real K-theory it gives the Pontrjagin character.

Of course, is $E$ is a ring spectrum so is $E_\mathbb{Q}$, and one must identify the induced ring structure on $H^*(X;\pi_{-*}(E)\otimes \mathbb{Q})$.