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Algebraic topology Definition

Real K-theory KO

Builds on Vector bundleGrothendieck groupReduced K-theory of a spaceComplex K-theory KUSuspensionCompact spaceHausdorff space

Definition

Let XX be a compact Hausdorff space. Consider the commutative monoid of isomorphism classes of real vector bundles over XX under direct sum. The group KO0(X)KO^0(X) is its Grothendieck group. For a pointed compact Hausdorff space (Y,y0)(Y, y_0), the reduced group KO~0(Y)\widetilde{KO}^0(Y) is the kernel of the restriction map KO0(Y)→KO0({y0})=ℤKO^0(Y) \to KO^0(\set{y_0}) = \Z, as for reduced complex K-theory. For n≥1n \ge 1, set KO−n(X)=KO~0(Σn(X+)),KO^{-n}(X) = \widetilde{KO}^0(\Sigma^n (X_+)), where X+X_+ is XX with a disjoint base point added and Σ\Sigma is the reduced suspension. Real K-theory KOKO is the family of functors X↦KO−n(X)X \mapsto KO^{-n}(X), n≥0n \ge 0.

Remarks

Example: KO−1(point)=KO~0(S1)≅ℤ/2KO^{-1}(\text{point}) = \widetilde{KO}^0(S^1) \cong \Z/2. A real line bundle over S1S^1 is either trivial or the Möbius bundle MM, and M⊕MM \oplus M is trivial. So [M]−1[M] - 1 generates KO~0(S1)\widetilde{KO}^0(S^1) and has order 22.

Non-example: complexification KO0(X)→K0(X)KO^0(X) \to K^0(X), [E]↦[E⊗ℝℂ][E] \mapsto [E \otimes_{\R} \C], is not injective in general. For X=S1X = S^1 it sends [M]−1≠0[M] - 1 \ne 0 to 00, because K~0(S1)=0\tilde{K}^0(S^1) = 0 in complex K-theory.

The groups KO−n(point)KO^{-n}(\text{point}) for n=0,1,…,7n = 0, 1, \ldots, 7 are ℤ,ℤ/2,ℤ/2,0,ℤ,0,0,0\Z, \Z/2, \Z/2, 0, \Z, 0, 0, 0, and they repeat with period 88. For compact XX, KO0(X)KO^0(X) is the set of homotopy classes of maps X→ℤ×BOX \to \Z \times BO.

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 spaceconceptReduced K-theory of a space, concept, no evidence levelReduced K-theory of a…conceptSuspension, concept, no evidence levelSuspensionconceptVector bundle, concept, no evidence levelVector bundleconceptReal K-theory KO, definition, no evidence levelReal K-theory KOdefinition
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. Reduced K-theory of a space concept
  6. Suspension concept
  7. Vector bundle concept
  8. Real K-theory KO definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_K-theory, Definitions
About this page
TagVK3Q16FXZQ
URIm2://VK3Q16FXZQ
Kinddefinition
WritingW0 stub
MSC55N15, 55R50
DefinesReal K-theory KO
UsesVector bundle, Grothendieck group, Reduced K-theory of a space, Complex K-theory KU, Suspension, Compact space, Hausdorff space
Changed3 November 2026

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