KR-theory

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title: "KR-theory" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["k-theory"] description: "Mathematics concept" topic_path: "general/k-theory" source: "https://en.wikipedia.org/wiki/KR-theory" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematics concept ::

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 = Rq + iRp (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

References

1. (1966). "K-theory and reality". The Quarterly Journal of Mathematics.

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