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.
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.
The $G$-equivariant $K$-theory of the point is the representation ring of the group $G$:
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.
The $G$-equivariant K-theory of the point for $G$ a compact Lie group is the representation ring of $G$.
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
See also at algebraic K-theory – References – On quotient stacks.
Introductions and surveys include
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)
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
Last revised on November 30, 2017 at 06:00:41. See the history of this page for a list of all contributions to it.