A general construction of partial Grothendieck transformations

Fulton and MacPherson introduced the notion of bivariant theories related to Riemann-Roch-theorems, especially in the context of singular spaces. This is powerful formalism, which is a simultaneous generalization of a pair of contravariant and covariant theories. Natural transformations of bivariant theories are called Grothendieck transformations, and these generalize a pair of ordinary natural transformations. But there are many situations, where such a bivariant theory or a corresponding Grothendieck transformation is only ”partially known”: characteristic classes of singular spaces (e.g. Stiefel-Whitney or Chern classes), cohomology operations (e.g. singular Adams Riemann-Roch and Steenrod operations for Chow groups) or equivariant theories (e.g. Lefschetz RiemannRoch). We introduce in this paper a simpler notion of partial (weak) bivariant theories and partial Grothendieck transformations, which applies to all these examples. Our main theorem shows, that a natural transformation of covariant theories, which commutes with exterior products, automatically extends uniquely to such a partial Grothendieck transformations of suitable partial (weak) bivariant theories ! In the above geometric situations one has for example to consider only morphisms, whose target is a smooth manifold, or more generally, a suitable ”homology manifold” (and in the general bivariant language this is related to the existence of suitable strong orientations). We illustrate our main theorem for the examples above, relating it to corresponding known Riemann-Roch theorems. Acknowledgements. This paper should be seen as a continuation of the basic work of W.Fulton and R.MacPherson [FM] about bivariant theories and Grothendieck transformations.