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rational equivariant K-theory

Contents

Context

Rational homotopy theory

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

While the rationalization of complex topological K-theory has lost most interesting information (being just a direct sum of ordinary cohomology rational cohomology in every even degree) the rationalization of its enhancement to equivariant cohomologyequivariant K-theory – is still a comparatively rich object, controlled by the rational representation theory of the given equivariance group.

Rational equivariant K-theory appears in a variety of different but equivalent concrete incarnations, such as Bredon cohomology with coefficients in the representation ring/class functions or as Chen-Ruan cohomology (here shown over the complex numbers, see Lück-Oliver 01 for discussion over the rational numbers):

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

As any Chern-Dold character, the equivariant Chern character on equivariant K-theory takes values in its rationalization. Hence, together with the above variety of presentations of rational equivariant K-theory, there is a corresponding variety of realizations of the equivariant Chern character.

References

For more see the references at equivariant Chern character.

Last revised on March 14, 2021 at 11:35:41. See the history of this page for a list of all contributions to it.