nLab delocalized equivariant cohomology

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

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

(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) $X \sslash G$. (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.

(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 theory	definition/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 cohomology	Baum-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-theory	Lück-Oliver 01, Thm. 5.5, Mislin-Valette 03, Thm. 6.1

Related concepts

• equivariant Chern character

References

• Paul Baum, Alain Connes, around (1.6) in: Chern character for discrete groups, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 (doi:10.1016/B978-0-12-480440-1.50015-0)

• Guido Mislin, Alain Valette, Theorem 6.1 in: Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (doi:10.1007/978-3-0348-8089-3)

• Ulrich Bunke, Markus Spitzweck, Thomas Schick, Inertia and delocalized twisted cohomology, Homotopy, Homology and Applications, vol 10(1), pp 129-180 (2008) (arXiv:math/0609576)

• Richard Szabo, Alessandro Valentino, Section 4.3 of: Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory, Commun. Math. Phys. 294:647-702, 2010 (arXiv:0710.2773)