I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that:
i) they vanish if the bundle of unit spheres $S(E)$ of the vector bundle $E$ (viewed as a real vector bundle if $E$ is complex) is stably fiberwise-homotopically trivial.
ii) the are well behaved under direct image(umkehr) homomorphism, for bundles oriented over the corresponding cohomology theory e.g., skew functoriality etc...
Other than Stiefel-Whitney and Wu classes in ordinary cohomology I know only Bott's cannibalistic classes in $K$-theory. But I don't know how Bott classes behave with regard to ii). Moreover my feeling is that, precisely because of ii), the right place for the characteristic classes which I need should be the complex cobordism.
Any reference to related topics would be more than welcome.