Roughly speaking, in cohomology theory, characteristic classes are elements of the cohomology of the base space of a fibre bundle which can tell you something about the nature of the fibre bundle. In "Characteristic Classes" by Milnor, he mentions that characteristic homology classes for the tangent bundle of a smooth manifold have been defined. I've tried looking these up without much luck. My question is: is there a simple, analogous notion of characteristic homology class to the regular one in cohomology? Would I be right in saying that these elements would live in the homology of the total space of a fibre bundle? Could they, for example, give obstruction to certain topological spaces being the base of a fibre bundle with the total space in question?