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KR-theory

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Mathematics concept

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 {\displaystyle \mathbb {C} } ×E to E commute with the involution, where the involution acts as complex conjugation on C {\displaystyle \mathbb {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 {\displaystyle \mathbb {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 = KR, where KR is suspension with respect to R = R + iR (with a switch in the order of p and q), given by

K R p , q ( X , Y ) = K R ( X × B p , q , X × S p , q ∪ Y × B p , q ) {\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, S are the unit ball and sphere in R.

References

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