KR-theory explained

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 ×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 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

KR ^{p,q} (X,Y) = KR(X x B ^{p,q} ,X x S ^{p,q} \cup Y x B ^{p,q} ) and B^p,q, S^p,q are the unit ball and sphere in R^p,q