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incarnations of rational equivariant topological K-theory -- table

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

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