## Does a “Chern character” exist for any generalized cohomology theory?

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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.

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I think, your question is related to the orientability of cohomology theories. There are many multiplicative transformations between bordism theories and other (co)homology theories such as oriented bordism to singular homology, spin bordism to K-theory, string bordism to TMF. As you see, the natural home for such a morphism is not always singular cohomology, but eg in the case of spin bordism you can, of course, compose the morphism to K-theory with the chern character. – Lennart Meier Jun 23 2010 at 8:39

## 2 Answers

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.

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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 `$H^*(X;\pi_{-*}(E)\otimes \mathbb{Q})$`.

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