A brief glance at K-theory

K-theory has its origins in the late 1950s generalization by Grothendieck of the Riemann-Roch theorem [2]. Algebraic geometry utterly baffles me, making me unfit to summarize Grothendieck’s achievement, but he appears to have associated to each X in some family of algebraic spaces a group K(X) that turns out to be a natural repository for information about X and various objects defined “over” X. Within this framework, he recovered classical Riemann-Roch as a special case of a general theorem involving K-groups. Hirzebruch and Atiyah soon realized that these ideas could be exported to the world of algebraic topology. The resulting K-theory of topological spaces— the subject of this paper— turns out to be quite powerful: among its early triumphs were Adams’ determination of the maximum number of linearly independent vector fields on S, and the formulation and proof of the Atiyah-Singer index theorem. (Further work by many others has extended the ideas and constructions of K-theory beyond topological spaces to C∗-algebras and more general rings, but we will not discuss this here.)