Bivariant theory

In mathematics, a bivariant theory was introduced by Fulton and MacPherson (Fulton & MacPherson 1981), 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”.

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.

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.)