group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
The Green-Julg theorem identifies, under some conditions, equivariant K-theory with operator K-theory of corresponding crossed product algebras.
The representation ring of $G$ over the complex numbers is the $G$-equivariant K-theory of the point, or equivalently by the Green-Julg theorem, if $G$ is a compact Lie group, the operator K-theory of the group algebra (the groupoid convolution algebra of the delooping groupoid of $G$):
The first isomorphism here follows immediately from the elementary definition of equivariant topological K-theory, since a $G$-equivariant vector bundle over the point is manifestly just a linear representation of $G$ on a complex vector space.
(e.g. Greenlees 05, section 3, Wilson 16, example 1.6 p. 3)
Under the identification (1) and the Atiyah-Segal completion map
one may ask for the Chern character of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual character of the representation $V$. 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 $V$ their first Chern class $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical isomorphism from 1-dimensional characters in $Hom_{Grp}(G,U(1))$ to the group cohomology $H^2_{grp}(G, \mathbb{Z})$ and further to the ordinary cohomology $H^2(B G, \mathbb{Z})$ of the classifying space $B G$:
More generally, for $n$-dimensional linear representations $V$ their first Chern class $c_1(\widehat V)$ is the previously defined first Chern-class of the line bundle $\widehat{\wedge^n V}$ corresponding to the $n$-th exterior power $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the determinant line bundle $det(\widehat{V}) = \widehat{\wedge^n V}$:
(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 $n$-dimensional representation $V$ is expressed in terms of its character $\chi_V$ as
For example, for a representation of dimension $n = 2$ this reduces to
(see also e.g. tom Dieck 09, p. 45)
$\,$
An isomorphism analogous to (1) identifies the $G$-representation ring over the real numbers with the equivariant orthogonal $K$-theory of the point in degree 0:
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:
Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-representations from equivariant vector bundles. This method is also called Dirac induction.
Specifically, applied to equivariant complex line bundles on coadjoint orbits of $G$, this is a K-theoretic formulation of the orbit method.
For $X$ a topological space equipped with a $G$-action for $G$ a topological group, write $X//G$ for the homotopy type of the corresponding homotopy quotient. A standard model for this is the Borel construction
The ordinary topological K-theory of $X//G$ is also called the Borel-equivariant K-theory of $X$, denoted
There is a canonical map
from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the Borel construction this is given by the composite
where the first map is pullback along the projection $X \times E G \to X$ and the first equivalence holds because the $G$-action on $X \times E G$ is free.
This map from genuine to Borel equivariant K-theory is not in general an isomorphism.
Specifically for $X$ the point, then $K_G(\ast) \simeq R(G)$ is the representation ring and $K_G^{Bor}(\ast) \simeq K(B G)$ is the topological K-theory of the classifying space $B G$ of $G$-principal bundles. In this case the above canonical map is of the form
This is never an isomorphism, unless $G$ is the trivial group. But the Atiyah-Segal completion theorem says that the map identifies $K(B G)$ as the completion of $R(G)$ at the ideal of virtual representations of rank 0.
(equivariant) cohomology | representing spectrum | equivariant cohomology of the point $\ast$ | cohomology of classifying space $B G$ |
---|---|---|---|
(equivariant) ordinary cohomology | HZ | Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$ | |
(equivariant) complex K-theory | KU | representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$ | Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$ |
(equivariant) complex cobordism cohomology | MU | $MU_G(\ast)$ | completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$ |
(equivariant) algebraic K-theory | $K \mathbb{F}_p$ | representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$ | Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$ |
(equivariant) stable cohomotopy | K $\mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S | Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$ | Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$ |
There is a 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
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
113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563
See also at algebraic K-theory – References – On quotient stacks.
Introductions and surveys include
John Greenlees, Equivariant version of real and complex connective K-theory, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (Euclid:1139839291)
N. C. Phillips, Equivariant K-theory for proper actions, Pitman Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989.
Bruce Blackadar, section 11 of K-Theory for Operator Algebras
Alexander Merkujev, Equivariant K-theory (pdf)
Zachary Maddock, An informal discourse on equivariant K-theory (pdf)
Dylan Wilson, Equivariant K-theory, 2016 (pdf, pdf)
The equivariant Chern character is discussed in
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
Discussion of combined twisted and equivariant and real K-theory
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:
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)
Last revised on March 31, 2019 at 14:33:56. See the history of this page for a list of all contributions to it.