cohomology

# 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 $X$ 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

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

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

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

There is a canonical map

$K_G(X) \to K_G^{Bor}(X)$

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

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

where the first map is pullback along the projection $X \times E G \to X$ 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

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

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.

### Relation to representation theory

The $G$-equivariant $K$-theory of the point is the representation ring of the group $G$:

$K_G(\ast) \simeq Rep(G) \,.$

Accordingly the construction of an index (push-forward to the point) in equivariant K-theory is a way of producing $G$-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.

## Examples

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

## 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

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

• 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 November 30, 2017 at 06:00:41. See the history of this page for a list of all contributions to it.