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 $$\Complex$$&times;E to E commute with the involution, where the involution acts as complex conjugation on $$\Complex$$. (This differs from the notion of a complex vector bundle in the category of Z/2Z spaces, where the involution acts trivially on $$\Complex$$.)

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 KRp,q = KRp+1,q+1, where KRp,q is suspension with respect to Rp,q = Rq + iRp (with a switch in the order of p and q), given by
 * $$KR^{p,q}(X,Y) = KR(X\times B^{p,q},X\times S^{p,q}\cup Y\times B^{p,q})$$

and Bp,q, Sp,q are the unit ball and sphere in Rp,q.