KK-theory

In mathematics, KK-theory is a common generalization both of K-homology and K-theory (more precisely operator K-theory), as an additive bivariant functor on separable C*-algebras. This notion was introduced by the Russian mathematician Gennadi Kasparov in 1980.
It was influenced by Atiyah's concept of Fredholm modules for the Atiyah–Singer index theorem, and the classification of extensions of C*-algebras by Brown–Douglas–Fillmore (Lawrence G. Brown, Ronald G. Douglas, Peter Arthur Fillmore 1977). In turn, it has had great success in operator algebraic formalism toward the index theory and the classification of nuclear C*-algebras, as it was the key to the solutions of many problems in operator K-theory, such as, for instance, the mere calculation of K-groups. Furthermore, it was essential in the development of the Baum–Connes conjecture and plays a crucial role in noncommutative topology.
KK-theory was followed by a series of similar bifunctor constructions...