KR-theory

In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by , motivated by applications to the Atiyah–Singer index theorem for real elliptic operators.
Definition.
A "real space" is a defined to be a topological space with an involution. A real vector bundle over a real space "X" is defined to be a complex vector bundle "E" over "X" that is also a real space, such that the natural maps from "E" to "X" and from C×"E" to "E" commute with the involution, where the involution acts as complex conjugation on C. (This differs from the notion of a complex vector bundle in the category of Z/2Z spaces, where the involution acts trivially on C.)
The group "KR"("X") is the Grothendieck group of finite-dimensional real vector bundles over the real space "X".
Periodicity.
Similarly to Bott periodicity, the periodicity theorem for "KR" states that "KR""p","q" = "KR""p"+1,"q"+1, where "KR""p","q" is suspension with respect to "R""p","q" =
R"q" + iR"p" (with a switch in the order of "p" and "q"), given by
and "B""p","q", "S""p","q" are the unit ball and sphere in "R""p","q".