Math 2.0

K-theory Definition

KR-theory

Builds on InvolutionVector bundleGrothendieck groupComplex K-theory KUReal K-theory KOCompact spaceHausdorff space

Definition

Let XX be a compact Hausdorff space with a continuous involution τ ⁣:X→X\tau \colon X \to X, so τ∘τ=id⁡\tau \circ \tau = \id. A Real vector bundle over (X,τ)(X, \tau) is a complex vector bundle p ⁣:E→Xp \colon E \to X with a continuous map σ ⁣:E→E\sigma \colon E \to E such that:

  1. σ∘σ=id⁡E\sigma \circ \sigma = \id_E and p∘σ=τ∘pp \circ \sigma = \tau \circ p;
  2. for each x∈Xx \in X, the restriction σ ⁣:Ex→Eτ(x)\sigma \colon E_x \to E_{\tau(x)} is antilinear: σ(λv)=λˉσ(v)\sigma(\lambda v) = \bar{\lambda} \sigma(v) for λ∈ℂ\lambda \in \C.

An isomorphism of Real vector bundles is an isomorphism of complex vector bundles that commutes with the maps σ\sigma. The group KR(X)KR(X) is the Grothendieck group of the monoid of isomorphism classes of Real vector bundles over (X,τ)(X, \tau) under direct sum.

Remarks

Example: let τ=id⁡X\tau = \id_X. For a Real bundle (E,σ)(E, \sigma), each σ\sigma is an antilinear involution of a fibre ExE_x, and its fixed points form a real vector bundle EσE^\sigma with E=Eσ⊗ℝℂE = E^\sigma \otimes_{\R} \C. So KR(X)≅KO(X)KR(X) \cong KO(X), the real K-theory of XX.

Example: let Y=X×{1,2}Y = X \times \set{1, 2} with τ\tau exchanging the two copies. A Real bundle over YY is determined by its restriction FF to the first copy, so KR(Y)≅K(X)KR(Y) \cong K(X), the complex K-theory of XX.

Non-example: if σ\sigma is complex linear on fibres instead of antilinear, clause 2 fails. Then (E,σ)(E, \sigma) is a bundle with an action of ℤ/2\Z/2 by linear maps; such bundles give equivariant K-theory, not KR-theory.

With ℝp,q=ℝq⊕iℝp\R^{p,q} = \R^q \oplus i \R^p and complex conjugation as involution, one defines groups KRp,q(X)KR^{p,q}(X) by suspension. Atiyah's (1,1)(1,1)-periodicity theorem states that suspension by ℝ1,1=ℂ\R^{1,1} = \C with conjugation does not change KR-theory; both complex and real Bott periodicity follow from it.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptComplex K-theory KU, concept, no evidence levelComplex K-theory KUconceptGrothendieck group, concept, no evidence levelGrothendieck groupconceptHausdorff space, concept, no evidence levelHausdorff spaceconceptInvolution, concept, no evidence levelInvolutionconceptReal K-theory KO, concept, no evidence levelReal K-theory KOconceptVector bundle, concept, no evidence levelVector bundleconceptKR-theory, definition, no evidence levelKR-theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Compact space concept
  2. Complex K-theory KU concept
  3. Grothendieck group concept
  4. Hausdorff space concept
  5. Involution concept
  6. Real K-theory KO concept
  7. Vector bundle concept
  8. KR-theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/KR-theory
About this page
TagPTYY4P13PE
URIm2://PTYY4P13PE
Kinddefinition
WritingW0 stub
MSC19L47, 55N15
DefinesKR-theory
UsesInvolution, Vector bundle, Grothendieck group, Complex K-theory KU, Real K-theory KO, Compact space, Hausdorff space
Changed3 November 2026

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