nLab
equivariant Chern character

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Rational homotopy theory

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
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

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Last revised on August 19, 2021 at 09:48:04. See the history of this page for a list of all contributions to it.