Co-operational bivariant theory (Q6585792)

Let \(\mathcal{C}\) be a category that has a final object \(pt\) and in which the fiber product is well-defined. Additionally, classes of ``confined maps'' and ``independent squares'' are distinguished. The main example is the category of complex algebraic varieties (or just pseudomanifolds), with the distinguished classes to be the proper maps and pull-back squares. The concept of bivariant theory was introduced in [\textit{W. Fulton} and \textit{R. MacPherson}, Categorical framework for the study of singular spaces. Providence, RI: American Mathematical Society (AMS) (1981; Zbl 0467.55005)], see also [\textit{S. Yokura}, Topology Appl. 123, No. 2, 283--296 (2002; Zbl 1045.55003)]. Every covariant functor, such as a homology theory \(h_*(-)\), gives rise to its bivariant counterpart: for a map \(X \to Y\), a graded abelian group \(\mathbb{B}^{op}h_*(X \to Y)\) is assigned. A dual notion, called ``co-operational'' bivariant theory, is defined axiomatically. Any cohomology theory defines such a co-operational theory. A transformation of multiplicative cohomology theories induces a transformation of co-operational bivariant theories. In particular, if \(h^*(-)=H^*(-)\) is the standard cohomology, then the co-operational theory contains information about both: cohomlogy \(H^*(X) = \mathbb{B}^{coop}H^*(X \to X)\) and the Borel-Moore homology, which is isomorphic to \(\mathbb{B}^{coop}H^*(X \to pt)\). Examples of theories such \(K\)-theory, Chow groups, Levine-Morel cobordisms and transformations: the Chern character, Adams operations are discussed.