KR-theory

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.

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

${\displaystyle KR^{p,q}(X,Y)=KR(X\times B^{p,q},X\times S^{p,q}\cup Y\times B^{p,q})}$

and B^p,q, S^p,q are the unit ball and sphere in R^p,q.

References

• Atiyah, Michael Francis (1966), "K-theory and reality", The Quarterly Journal of Mathematics. Oxford. Second Series, 17 (1): 367–386, doi:10.1093/qmath/17.1.367, ISSN 0033-5606, MR 0206940