北京雁栖湖应用数学研究院

K-theory, as the study of stable isomorphism classes of vector bundles, originated in the work of Grothendieck in 1950's as part of his programme of revolutionising algebraic geometry. In later work, it was turned into a "universal" cohomology theory, indispensable for the study of both commutative and non-commutative algebras, starting with the work of Bott, Atiyah, and many others. The dual homology theory, the K-homology, came relatively soon afterwards, its geometric picture essentially due to Atiyah. In this picture, elliptic differential operators on vector bundles and their indices played a prominent role.

Parallel to this, K-homology appeared in the work of Brown, Douglas, and Fillmore who studied a completely different question. The problem was as follows. Given a bounded operator A on a Hilbert space which is almost normal, i.e. A*A - AA* is a compact operator, under what conditions does there exist a normal operator B such that A-B is a compact operator. The most natural language for this kind of problem is that of equivalence classes of extensions. As it turned out, the involved extension theory was "just" another version of K-homology (for C*-algebras).

The C*-algebras entered naturally in the picture via Grothendieck's basic philosophy, which says that a study of a space M with some structure (smooth, analytic, algebraic, measurable) is equivalent to the study of the corresponding algebra of functions A = C#(M), where # refers to the structure in question. It is quite natural to use the language of non-commutative geometry, where the algebra A does not have to be commutative and hence the notion of "points" does not survive. The C*-structure is, to a degree, a generalisation of the algebra complex valued continuous functions on a compact space or, at the other extreme, of the algebra of compact operators on a separable Hilbert space appearing in quantum mechanics.
Finally, in the hands of Kasparov, the K-theory and K-homology became a bi-variant functor, the KK-theory. While the existence of a bivariant theory is, from a purely topological/homological algebra point of view, quite natural, the Kasparov approach is based on the construction of KK-cycles, which somehow mirror elliptic operators in the non-commutative algebra context and satisfies remarkable properties which make it a very versatile tool in many situations.

Kasparov's original motivation was the Novikov conjecture, which since then has become the main subject of coarse geometry. However, many computations in K-theory and classification theory of C*-algebras and group actions still rest on the full power of KK-theory.