# nLab delocalized equivariant cohomology

Contents

cohomology

### Theorems

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Let $G$ be a finite group and $X$ a suitable topological G-space. Then the delocalized $G$-equivariant cohomology of $X$ (Connes-Baum 89, p. 165) is the ordinary cohomology of the quotient space

$\left( \underset{ h \in G }{\sqcup} X^{h} \right) / G$

of the disjoint union of fixed loci, where the action of $g \in G$ is by

$\left(h,\, x \in X^h \right) \;\; \overset{g}{\mapsto} \;\; \left(g^{-1}\cdot h \cdot g ,\, x \cdot g \in X^{g^{-1} h g} \right) \,.$

## Properties

### Relation to Chen-Ruan cohomology

The definition of delocalized equivariant cohomology coincides with that of Chen-Ruan cohomology for the global quotient orbifold (orbispace) $X \sslash G$. (Manifestly so, also highlighted in Bunke-Spitzweck-Schick 06, Section 1.3)

### Relation to Bredon cohomology

In turn, Chen-Ruan cohomology of global quotient orbifolds (and hence, by the above delocalized equivariant cohomology) coincides with Bredon cohomology with coefficients in the rationalized representation ring-functor $G/H \mapsto Rep(H)$ on the orbit category.

### Relation to rationalized equivariant K-theory

In further turn, even-periodic Bredon cohomology with coefficients in the representation ring-functor is equivalent to rationalized equivariant K-theory:

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

Last revised on August 18, 2021 at 12:54:55. See the history of this page for a list of all contributions to it.