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equivariant K-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Equivariant K-theory is the equivariant cohomology version of the generalized cohomology theory K-theory.

To the extent that K-theory is given by equivalence classes of virtual vector bundles (topological K-theory, operator K-theory), equivariant K-theory is given by equivalence classes of virtual equivariant bundles or generalizations to noncommutative topology thereof, as in equivariant operator K-theory, equivariant KK-theory.

Properties

Relation to operator K-theory of crossed product algebras

The Green-Julg theorem identifies, under some conditions, equivariant K-theory with operator K-theory of corresponding crossed product algebras.

Relation to K-theory of homotopy quotient spaces (Borel constructions)

For XX a topological space equipped with a GG-action for GG a topological group, write X//GX//G for the homotopy type of the corresponding homotopy quotient. A standard model for this is the Borel construction

X//G(X×EG)/G. X//G \simeq (X \times EG)/G \,.

The ordinary topological K-theory of X//GX//G is also called the Borel-equivariant K-theory of XX, denoted

K G Bor(X)K(X//G). K_G^{Bor}(X) \coloneqq K(X//G) \,.

There is a canonical map

K G(X)K G Bor(X) K_G(X) \to K_G^{Bor}(X)

from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the Borel construction this is given by the composite

K G(X)K G(X×EG)K((X×EG)/G)K G Bor(X), K_G(X) \to K_G(X \times E G) \simeq K((X \times E G) / G ) \simeq K_G^{Bor}(X) \,,

where the first map is pullback along the projection X×EGXX \times E G \to X and the first equivalence holds because the GG-action on X×EGX \times E G is free.

This map from genuine to Borel equivariant K-theory is not in general an isomorphism.

Specifically for XX the point, then K G(*)R(G)K_G(\ast) \simeq R(G) is the representation ring and K G Bor(*)K(BG)K_G^{Bor}(\ast) \simeq K(B G) is the topological K-theory of the classifying space BGB G of GG-principal bundles. In this case the above canonical map is of the form

R(G)K(BG). R(G) \to K(B G) \,.

This is never an isomorphism, unless GG is the trivial group. But the Atiyah-Segal completion theorem says that the map identifies K(BG)K(B G) as the completion of R(G)R(G) at the ideal of virtual representations of rank 0.

Relation to representation theory

The GG-equivariant KK-theory of the point is the representation ring of the group GG:

K G(*)Rep(G). K_G(\ast) \simeq Rep(G) \,.

Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing GG-representations from equivariant vector bundles. This method is also called Dirac induction.

Specifically, applied to equivariant complex line bundles on coadjoint orbits of GG, this is a K-theoretic formulation of the orbit method.

Examples

The GG-equivariant K-theory of the point for GG a compact Lie group is the representation ring of GG.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)𝕂𝕌 G(*)compl.𝕂𝕌 G(*)^𝕂𝕌(BG)R(G) \simeq \mathbb{KU}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{KU}_G(\ast)} \simeq \mathbb{KU}(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
(equivariant)
stable cohomotopy
K 𝔽 1Segal 74\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)

References

The idea of equivariant topological K-theory goes back to

  • Graeme Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151

  • Graeme Segal, Michael Atiyah, Equivariant K-theory and completion, J. Differential Geometry 3 (1969), 1–18. MR 0259946 (41 #4575

and for algebraic K-theory to

  • Robert Thomason, Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol.

    113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563

See also at algebraic K-theory – References – On quotient stacks.

Introductions and surveys include

Discussion relating to K-theory of homotopy quotients/Borel constructions is in

Discussion of the adjoint action-equivariant K-theory of suitable Lie groups in in

Discussion of K-theory of orbifolds is for instance in section 3 of

  • Alejandro Adem, Johanna Leida, Yongbin Ruan, Orbifolds and string topology, Cambridge Tracts in Mathematics 171, 2007 (pdf)

Discussion of combined twisted and equivariant and real K-theory

Last revised on September 10, 2018 at 12:27:03. See the history of this page for a list of all contributions to it.