# Bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton 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 $\displaystyle{ f : X \to Y }$ be a map. For such a map, we can consider the fiber square

$\displaystyle{ \begin{matrix} X' & \to & Y' \\ \downarrow & & \downarrow \\ X & \to & Y \end{matrix} }$

(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 $\displaystyle{ f }$.

Now, a birational class of $\displaystyle{ f }$ is a family of group homomorphisms indexed by the fiber squares:

$\displaystyle{ A_k Y' \to A_{k-p} X' }$

satisfying the certain compatibility conditions.

## Operational Chow ring

The basic question was whether there is a cycle map:

$\displaystyle{ A^*(X) \to \operatorname{H}^*(X, \mathbb{Z}). }$

If X is smooth, such a map exists since $\displaystyle{ A^*(X) }$ is the usual Chow ring of X. (Totaro 2014) 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.)