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In mathematics, '''KR-theory''' is a variant of [[topological K-theory]] defined for spaces with an [[involution (mathematics)|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 '''C'''&times;''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'''/2'''Z''' 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''<sup>''p'',''q''</sup> = ''KR''<sup>''p''+1,''q''+1</sup>, where ''KR''<sup>''p'',''q''</sup> is suspension with respect to ''R''<sup>''p'',''q''</sup> =
'''R'''<sup>''q''</sup> + i'''R'''<sup>''p''</sup> (with a switch in the order of ''p'' and ''q''), given by
:<math>KR^{p,q}(X,Y) = KR(X\times B^{p,q},X\times S^{p,q}\cup Y\times B^{p,q})</math>
and ''B''<sup>''p'',''q''</sup>, ''S''<sup>''p'',''q''</sup> are the unit ball and sphere  in ''R''<sup>''p'',''q''</sup>.

==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 | mr=0206940 | year=1966 | journal=The Quarterly Journal of Mathematics. Oxford. Second Series | issn=0033-5606 | volume=17 | pages=367–386 | doi=10.1093/qmath/17.1.367 | issue=1}}

[[Category:K-theory]]