Contents

cohomology

# Contents

## Idea

Equivariant ordinary cohomology is the equivariant cohomology-version of ordinary cohomology.

By default this is often understood to be the Borel equivariant cohomology-version of ordinary cohomology, but more generally this must be understood as Bredon cohomology, i.e. as the proper-equivariant version of ordinary cohomology.

## Properties

### Equivariant Chern character

There is a Chern character map from equivariant K-theory to equivariant ordinary cohomology.

(e.g. Stefanich)

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## References

### As Bredon-equivariant cohomology theory

Discussion over the point but in arbitrary RO(G)-degree:

for equivariance group a dihedral group of order $2p$:

• Yunze Lu, On the $RO(G)$-graded coefficients of $Q_8$ equivariant cohomology, Topology and its Applications, 2021 (doi:10.1016/j.topol.2021.107921)

Last revised on December 22, 2021 at 10:56:35. See the history of this page for a list of all contributions to it.