# nLab equivariant Chern character

Contents

cohomology

### Theorems

#### Representation theory

representation theory

geometric representation theory

and

# Contents

## Idea

The equivariant Chern character is the generalization of the Chern character from topological K-theory to equivariant topological K-theory, equivalently the specialization of the equivariant Chern-Dold character for generalized equivariant cohomology to equivariant topological K-theory.

The equivariant Chern character has a variety of different but equivalent concrete incarnations, depending on the choice of presentation of the rational equivariant K-theory that it takes values in:

Incarnations of rational equivariant K-theory:

cohomology theorydefinition/equivalence due to
$\simeq K_G^0\big(X; \mathbb{C} \big)$rational equivariant K-theory
$\simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big)$delocalized equivariant cohomologyBaum-Connes 89, Thm. 1.19
$\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)$Chen-Ruan cohomology
of global quotient orbifold
Chen-Ruan 00, Sec. 3.1
$\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)$Bredon cohomology
with coefficients in representation ring
Ho88 6.5+Ho90 5.5+Mo02 p. 18,
Mislin-Valette 03, Thm. 6.1,
Szabo-Valentino 07, Sec. 4.2
$\simeq K_G^0\big(X; \mathbb{C} \big)$rational equivariant K-theoryLück-Oliver 01, Thm. 5.5,
Mislin-Valette 03, Thm. 6.1

## References

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Last revised on December 20, 2021 at 05:14:47. See the history of this page for a list of all contributions to it.