nLab
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 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):

(1)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)

Under this identification 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(BG) expressed in terms of the actual character of the representation VV. 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 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, ite, (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

(2)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 (1) 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{H}}(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
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(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 KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
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)
(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)

Equivariant Chern-character

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

(e.g. Stefanich)

References

General

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

  • Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam)

  • Michael Atiyah, Friedrich Hirzebruch, Vector bundle and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 (pdf)

  • 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

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

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 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

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.

Last revised on October 20, 2018 at 07:09:25. See the history of this page for a list of all contributions to it.