Real K-theory

Real K-theory, denoted KO(X), is defined for a compact Hausdorff space X with a continuous involution. It classifies real vector bundles over X that are equipped with an involution lifting the one on X, with the involution acting anti-linearly on fibers. The group KO(X) is the Grothendieck group of such bundles, and it fits into a long exact sequence relating it to ordinary K-theory and to the K-theory of the fixed-point set.¹

Real K-theory exhibits a periodicity of order 8, known as Bott periodicity, which is a consequence of the Bott periodicity theorem for real vector bundles. The coefficient groups KO_n(point) are periodic with period 8, and they are given by the homotopy groups of the real Bott spectrum. This periodicity is fundamental in computations and in the stable homotopy category.²

Real K-theory is used to determine the maximum number of linearly independent vector fields on spheres, a problem solved by Adams using KO-theory. It also appears in the Atiyah–Singer index theorem for real elliptic operators, and in the study of real C*-algebras and their K-theory. In mathematical physics, it underlies the classification of topological phases of matter, where the 8-fold periodicity matches the periodic table of topological insulators.³

Real K-theory is often confused with KR-theory, which is a different theory for spaces with an involution, but the two are related by a periodicity of order 8. A lesser-known fact is that the notation KO is also used for the K-theory of real vector bundles without involution, which can cause ambiguity. The theory has connections to the classification of real Clifford algebras, and it appears in the solution of the Hopf invariant one problem. Additionally, the equivariant version, where the involution is part of a group action, has applications in equivariant stable homotopy theory.⁴

Involution

A map that is its own inverse, often used to define symmetries.

Grothendieck group

A group constructed from a commutative monoid, used to formalize the notion of virtual differences.

Bott periodicity

A periodicity theorem for the homotopy groups of classical groups, leading to periodicity in K-theory.