Math 2.0

K-theory Definition

Equivariant K-theory

Builds on Compact Lie groupGroup actionVector bundleGrothendieck groupGrothendieck group of vector bundlesRepresentation ringCompact space

Definition

Let GG be a compact Lie group and let XX be a compact Hausdorff space with a continuous action of GG. A GG-vector bundle over XX is a complex vector bundle p ⁣:E→Xp \colon E \to X with a continuous action of GG on EE such that:

  1. p(g⋅v)=g⋅p(v)p(g \cdot v) = g \cdot p(v) for all g∈Gg \in G and v∈Ev \in E;
  2. for each g∈Gg \in G and x∈Xx \in X, the map Ex→EgxE_x \to E_{g x}, v↦g⋅vv \mapsto g \cdot v, is complex linear.

An isomorphism of GG-vector bundles is a bundle isomorphism that commutes with the action of GG. The equivariant K-theory KG(X)K_G(X) is the Grothendieck group of the monoid of isomorphism classes of GG-vector bundles over XX under direct sum.

Remarks

The tensor product makes KG(X)K_G(X) a commutative ring, and an algebra over KG(point)K_G(\text{point}). For trivial GG it is the Grothendieck group of vector bundles K(X)K(X).

Example: for XX a point, a GG-vector bundle is a finite-dimensional complex representation of GG, so KG(point)=R(G)K_G(\text{point}) = R(G), the representation ring of GG.

Example: if GG acts freely on XX, pulling back along X→X/GX \to X/G gives an isomorphism K(X/G)≅KG(X)K(X/G) \cong K_G(X).

Non-example: let G=ℤ/2G = \Z/2 act on a vector bundle by a map that is antilinear on fibres, such as complex conjugation on X×ℂX \times \C, with GG fixing every point of XX. Clause 2 fails, so this is not a GG-vector bundle; such objects give KR-theory instead.

Groups KG−n(X)K^{-n}_G(X) are defined from suspensions with trivial action, as in the non-equivariant case, and they satisfy Bott periodicity of period 22.

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 Lie group, concept, no evidence levelCompact Lie groupconceptCompact space, concept, no evidence levelCompact spaceconceptGrothendieck group, concept, no evidence levelGrothendieck groupconceptGrothendieck group of vector bundles, concept, no evidence levelGrothendieck group of…conceptGroup action, concept, no evidence levelGroup actionconceptRepresentation ring, concept, no evidence levelRepresentation ringconceptVector bundle, concept, no evidence levelVector bundleconceptEquivariant K-theory, definition, no evidence levelEquivariant K-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.
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  1. Compact Lie group concept
  2. Compact space concept
  3. Grothendieck group concept
  4. Grothendieck group of vector bundles concept
  5. Group action concept
  6. Representation ring concept
  7. Vector bundle concept
  8. Equivariant K-theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Topological_K-theory
About this page
Tag8DP8SZ56EG
URIm2://8DP8SZ56EG
Kinddefinition
WritingW0 stub
MSC19L47, 55N91
DefinesEquivariant K-theory
UsesCompact Lie group, Group action, Vector bundle, Grothendieck group, Grothendieck group of vector bundles, Representation ring, Compact space
Changed3 November 2026

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