equivariant K-theory





Special and general types

Special notions


Extra structure



Representation theory



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.


Bott periodicity

The Bott periodicity of plain K-theory generalizes to equivariant K-theory:

Complex equivariant K-theory is invariant under smashing with representation spheres of complex representations (Atiyah 68, Theorem 4.3), while real equivariant K-theory is invariant under smashing with representation spheres of real 8d reps with spin structure (Atiyah 68, Theorem 6.1).

Review in Karoubi 05, Section 5.

Complex orientation

Equivariant complex K-theory is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 10).


(equivariant K-theory of projective G-space)

For GG an abelian compact Lie group, let

i1 V iGRepresentations fin \underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin}

be a finite-dimensional direct sum of complex 1-dimensional linear representations.

The GG-equivariant K-theory ring K G()K_G(-) of the corresponding projective G-space P()P(-) is the following quotient ring of the polynomial ring in one variable LL over the complex representation ring R(G)R(G) of GG:

(1)K G(P(i1 V i))R(G)[L]/i(11 V iL), K_G \Big( P \big( \underset{i}{\oplus} \, \mathbf{1}_{V_i} \big) \Big) \;\; \simeq \;\; R(G) \big[ L \big] \big/ \underset{i}{\prod} \big( 1 - 1_{{}_{V_i}} L \big) \,,


(Greenlees 01, p. 248 (24 of 39))


(equivariant complex orientation of equivariant K-theory)

For GG an abelian compact Lie group and 1 VGRepresentations fin\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin} a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space S 1 VP(1 V1)S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big) (this Prop.) and so, by Prop. ,

K˜ G(S 1 V) K G(P(1 V1);P(1)pt) ker(R(G)[L]/(11 VL)(1L)R(G)[L]/(1L)R(G)) (1L)R(G)[L]/(11 VL)(1L) \begin{aligned} \widetilde K_G \big( S^{\mathbf{1}_V} \big) & \simeq\, K_G \big( P( \mathbf{1}_V \oplus \mathbf{1} ) ; \, \underset{ \simeq \, pt }{ \underbrace{ P( \mathbf{1} ) } } \big) \\ & \simeq ker \Big( R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \longrightarrow \underset{ \simeq \, R(G) }{ \underbrace{ R(G)\big[L\big] \big/ (1 - L) } } \Big) \\ & \simeq (1 - L) \cdot R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \end{aligned}

is generated by the Bott element (1L)(1 - L) over P(1 V1)P\big( \mathbf{1}_V \oplus \mathbf{1} \big). By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space P(𝒰 G)P\big(\mathcal{U}_G\big). The latter is thereby exhibited as an equivariant complex orientation in equivariant complex K-theory.

(Greenlees 01, p. 248 (24 of 39))

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 representation theory

Equivariant KUKU and the complex representation ring

The representation ring of GG over the complex numbers is the GG-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if GG is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of GG):

(2)R (G)KU G 0(*)KK(,C(BG)). R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,.

The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a GG-equivariant vector bundle over the point is manifestly just a linear representation of GG on a complex vector space.

(e.g. Greenlees 05, section 3, Wilson 16, example 1.6 p. 3)

Chern classes of linear representations

Under the identification (2) and the Atiyah-Segal completion map

R (G)KU G 0(*)()^KU(BG) R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \overset{ \widehat{(-)} }{\longrightarrow} KU(BG)

one may ask for the Chern character of the K-theory class V^KU(BG)\widehat{V} \in KU(B G) expressed in terms of the actual character of the representation VV. For more see at Chern class of a linear representation.

There is a closed formula at least for the first Chern class (Atiyah 61, appendix):

For 1-dimensional representations VV their first Chern class c 1(V^)H 2(BG,)c_1(\widehat{V}) \in H^2(B G, \mathbb{Z}) is their image under the canonical isomorphism from 1-dimensional characters in Hom Grp(G,U(1))Hom_{Grp}(G,U(1)) to the group cohomology H grp 2(G,)H^2_{grp}(G, \mathbb{Z}) and further to the ordinary cohomology H 2(BG,)H^2(B G, \mathbb{Z}) of the classifying space BGB G:

c 1(()^):Hom Grp(G,U(1))H grp 2(G,)H 2(BG,). c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,.

More generally, for nn-dimensional linear representations VV their first Chern class c 1(V^)c_1(\widehat V) is the previously defined first Chern-class of the line bundle nV^\widehat{\wedge^n V} corresponding to the nn-th exterior power nV\wedge^n V of VV. The latter is a 1-dimensional representation, corresponding to the determinant line bundle det(V^)= nV^det(\widehat{V}) = \widehat{\wedge^n V}:

c 1(V^)=c 1(det(V^))=c 1( nV^). c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,.

(Atiyah 61, appendix, item (7))

More explicitly, via the formula for the determinant as a polynomial in traces of powers (see there) this means that the first Chern class of the nn-dimensional representation VV is expressed in terms of its character χ V\chi_V as

(3)c 1(V)=χ ( nV):gk 1,,k n=1nk =nl=1n(1) k l+1l k lk l!(χ V(g l)) k l c_1(V) = \chi_{\left(\wedge^n V\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l}

For example, for a representation of dimension n=2n = 2 this reduces to

c 1(V)=χ VV:g12((χ V(g)) 2χ V(g 2)) c_1(V) = \chi_{V \wedge V} \;\colon\; g \;\mapsto\; \frac{1}{2} \left( \left( \chi_V(g)\right)^2 - \chi_V(g^2) \right)

(see also e.g. tom Dieck 09, p. 45)


Equivariant KOKO and the real representation ring

An isomorphism analogous to (2) identifies the GG-representation ring over the real numbers with the equivariant orthogonal KK-theory of the point in degree 0:

R (G)KO G 0(*). R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,.

But beware that equivariant KO, even of the point, is much richer in higher degree (Wilson 16, remark 3.34).

In fact, equivariant KO-theory of the point subsumes the representation rings over the real numbers, the complex numbers and the quaternions:

KO G n(*){0 | n=7 R (G)/R (G) | n=6 R (G)/R (G) | n=5 R (G)/R (G) | n=4 0 | n=3 R (G)/R (G) | n=2 R (G)/R (G) | n=1 R (G)/R (G) | n=0 KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right.

(Greenlees 05, p. 3)

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.

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.

(equivariant) cohomologyrepresenting
equivariant cohomology
of the point *\ast
of classifying space BGB G
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
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 KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
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="">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
stable cohomotopy
K𝔽 1Segal 74K \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="">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)


Incarnations of rational equivariant K-theory:

cohomology theorydefinition/equivalence due to
H ev((hHX h)/G;) \phantom{\simeq} H^{ev}\Big( \big(\underset{h \in H}{\coprod} X^h\big)/G; \mathbb{C} \Big) delocalized equivariant cohomologyConnes-Baum 89
H CR ev((XG);)\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)Chen-Ruan cohomology
of global quotient orbifold
Chen-Ruan 00
H G ev(X;(G/HRep(H)))\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)Bredon cohomology
with coefficients in representation ring
Honkasalu 90, Thm. 5.5
Moerdijk 02, p. 18
K G 0(X;)\simeq K_G^0\big(X; \mathbb{C} \big) rational equivariant K-theoryLück-Oliver 01, Thm. 5.5

Equivariant Chern-character

There is an equivariant Chern character map from equivariant K-theory to equivariant ordinary cohomology.

(e.g. Stefanich)



The idea of equivariant topological K-theory and the Atiyah-Segal completion theorem goes back to

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

with construction via permutative categories in

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

Introductions and surveys:

On Bott periodicity in equivariant K-theory:

Basic computations:

  • Yimin Yang, On the Coefficient Groups of Equivariant K-Theory, Transactions of the American Mathematical Society

    Vol. 347, No. 1 (Jan., 1995), pp. 77-98 (jstor:2154789)

  • Max Karoubi, Equivariant K-theory of real vector spaces and real vector bundles, Topology and its Applications, 122, (2002) 531-456 (arXiv:math/0509497)

The equivariant Chern character is discussed in

  • German Stefanich, Chern Character in Twisted and Equivariant K-Theory (pdf)

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

Discussion of the twisted ad-equivariant K-theory of compact Lie groups:

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

Discussion of differential K-theory of orbifolds:

Discussion of combined twisted and equivariant and real K-theory

Discussion in the context of equivariant complex oriented cohomology theory:

Representing equivariant spectrum

That GG-equivariant topological K-theory is represented by a topological G-space is

This is enhanced to a representing naive G-spectrum in

Review includes:

  • Valentin Zakharevich, Section 2.2 of: K-Theoretic Computation of the Verlinde Ring (pdf)

  • Michael L. Ortiz, Theorem 2.2 in: Differential Equivariant K-Theory (arXiv:0905.0476)

In its incarnation (under Elmendorf's theorem) as a Spectra-valued presheaf on the GG-orbit category this is discussed in

  • James Davis, Wolfgang Lück, Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory, K-Theory 15:201–252, 1998 (pdf)

For D-brane charge on orbifolds

The proposal that D-brane charge on orbifolds is given by equivariant K-theory goes back to

but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in

For further references see at fractional D-brane.

On Chern classes of linear representations:

  • Atiyah 61, Appendix

  • Leonard Evens, On the Chern Classes of Representations of Finite Groups, Transactions of the American Mathematical Society Vol. 115 (Mar., 1965), pp. 180-193 (doi:10.2307/1994264)

  • F. Kamber, Ph. Tondeur, Flat Bundles and Characteristic Classes of Group-Representations, American Journal of Mathematics Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (doi:10.2307/2373408)

  • Peter Symonds, A splitting principle for group representations, Comment. Math. Helv. (1991) 66: 169 (doi:10.1007/BF02566643)

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