KR-theory

In mathematics, KR-theory is a variant of topological K-theory defined for spaces with an involution. It was introduced by {{harvtxt|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 $\Complex$ ×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

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

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

References

{{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | title=K-theory and reality | url=http://qjmath.oxfordjournals.org/cgi/reprint/17/1/367 | archive-url=https://archive.today/20130415131729/http://qjmath.oxfordjournals.org/cgi/reprint/17/1/367 | url-status=dead | archive-date=2013-04-15 | mr=0206940 | year=1966 | journal=The Quarterly Journal of Mathematics |series=Second Series | issn=0033-5606 | volume=17 | pages=367–386 | doi=10.1093/qmath/17.1.367 | issue=1| url-access=subscription }}

Category:K-theory