nLab
delocalized equivariant cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

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

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

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

(h,xX h)g(g 1hg,xgX g 1hg). \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) \,.

(Baum-Connes 89, above (7.3))

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) XGX \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/HRep(H)G/H \mapsto Rep(H) on the orbit category.

(Mislin-Valette 03, Thm. 6.1, review in Szabo-Valentino 07, Sec. 4.3

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
K G 0(X;)\simeq K_G^0\big(X; \mathbb{C} \big) rational equivariant K-theory
H ev((gGX g)/G;) \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
H CR ev((XG);)\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
H G ev(X;(G/HRep(H)))\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
K G 0(X;)\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.