  Fulton and MacPherson introduced the notion of bivariant theories and
Grothendieck transformations related to Riemann-Roch-theorems. 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 Riemann-Roch). 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 transformation 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.

Algebraic Geometry;Algebraic Topology