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 $$f : X \to Y$$ be a map. For such a map, we can consider the fiber square

\begin{matrix} X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end{matrix} $$ (for example, a blow-up.) Intuitively, the consideration of all the fiber squares like the above can be thought of as an approximation of the map $$f$$.

Now, a birational class of $$f$$ is a family of group homomorphisms indexed by the fiber squares:
 * $$A_k Y' \to A_{k-p} X'$$

satisfying the certain compatibility conditions.

Operational Chow ring
The basic question was whether there is a cycle map:
 * $$A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}).$$

If X is smooth, such a map exists since $$A^*(X)$$ 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.)