In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution.
It was introduced by Atiyah (1966), motivated by applications to the Atiyah–Singer index theorem for real elliptic operators.
(This differs from the notion of a complex vector bundle in the category of Z/2Z spaces, where the involution acts trivially on
The group KR(X) is the Grothendieck group of finite-dimensional real vector bundles over the real space X.
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 and Bp,q, Sp,q are the unit ball and sphere in Rp,q.