vap: My understanding is that wrong way (pushforward) maps on topological K-theory will only be defined under additional assumptions such as spaces being smooth and compact (or at least the space on the top is smooth, and the map is proper). Otherwise pushforward of a vector bundle does not have to be a finite complex of vector bundles. At least that's how it is in algebraic K-theory: K-theory is contravariantly functorial for all morphisms, and G-theory is covariantly functorial for proper morphisms.

მამუკა ჯიბლაძე: for the Baum-Douglas cycles, is the basic assumption that $X$ is a manifold? Would then K-cohomology and K-homology canonically isomorphic (in the compact manifold case)?

user36931: I looked at the paper of Thomason, and I don't quite yet see how it's relevant. Do you mean his approach of inverting the Bott element in the Algebraic K-theory to relate to topological K-theory?