The Chern character is a ring homomorphism from the complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology theory to the usual cohomlogy. Are they related with Atiyah-Hirzebruch?
2) And if there are such nice homomorphisms, what is the "Todd genus" in these cases, making the generalization of that famous diagram in Grothendieck–Hirzebruch–Riemann–Roch commute?
When I think about it, I cannot even recall seeing anything like this in real K-theory, but that is probably because I dont really know real K-theory at all.
Thank you very much.