Does a "Chern character" exist for any generalized cohomology theory? - MathOverflow

Does a "Chern character" exist for any generalized cohomology theory?

2010-06-23T07:55:03Z

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.

Answer by Oscar Randal-Williams for Does a "Chern character" exist for any generalized cohomology theory?

2010-06-23T08:53:57Z

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

Answer by Tom Goodwillie for Does a "Chern character" exist for any generalized cohomology theory?

2010-06-23T11:21:33Z

In Oscar Randall-Williams' answer "connective" is unnecessary. Also, there is no need to choose that isomorphism (from a rational theory to the ordinary theory with the same coefficient groups); it is canonical.

The generalization of the Todd genus or Todd class arises when the multiplicative theory $E$ has a "complex orientation": a multiplicatively well-behaved way of producing Thom isomorphisms in $E^*$-theory for all complex vector bundles.