at.algebraic topology - Characteristic classes in generalized cohomology theories?

Not a real answer to your question, but I think it may be related. One can ask the same question fofr Chern classes. For some generalized cohomology theories you can define Chern classes of complex vector bundles, and these satisfy the usual axioms for Chern classes.

What changes is the tensor product behviour. Given two line bundles L and M, the first Chern class $c_1(L otimes M)$ is given by a universal power series in $c_1(L)$ and $c_1(M)$. For instance in ordinary cohomology $c_1(L otimes M) = c_1(L) + c_1(M)$, while in K theory $c_1(L otimes M) = c_1(L) cdot c_1(M)$. Since line bundles form a group under tensor product, this power series is a (1-dimensional) formal group law.

So to any such cohomology theory you can attach a 1-dimensional formal group. For instance you attach the additive group to ordinary cohomology and the multiplicative group to K-theory. It turns out that you can go the other way round. The other 1-dimensional formal group laws are formal expansions of the group law of an elliptic curve near the origin; these give rise to the so called elliptic cohomology theories.

I think you can find details on the above constructions in any reference about elliptic cohomology. I've only heard about these theories, so I don't know an good reference. By the way, since I'm not an expert, please correct me if anything I have written above is wrong.