Bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson , in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring.
On technical levels, a bivariant theory is a mix of a homology theory and a cohomology theory. In general, a homology theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the category of (nice) spaces to the category of rings. A bivariant theory is a functor both covariant and contravariant; hence, the name “bivariant”.
Definition.
Unlike a homology theory or a cohomology theory, a bivariant class is defined for a map not a space.
Let formula_1 be a map. For such a map, we can consider the fiber square
(for example, a blow-up.) Intuitively, the consideration of all the fiber squares like the above can be thought of as an approximation of the map formula_3.
Now, a birational class of formula_3 is a family of group homomorphisms indexed by the fiber squares:
satisfying the certain compatibility conditions.
Operational Chow ring.
The basic question was whether there is a cycle map:
If "X" is smooth, such a map exists since formula_7 is the usual Chow ring of "X". has shown that rationally there is no such a map with good properties even if "X" is a linear variety, roughly a variety admitting a cell decomposition. He also notes that Voevodsky's motivic cohomology ring is "probably more useful " than the operational Chow ring for a singular scheme (§ 8 of loc. cit.)