nLab equivariant ordinary cohomology

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)

Related concepts

• equivariant de Rham cohomology

• equivariant ordinary differential cohomology

(equivariant) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation 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 $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

References

As Borel-equivariant cohomology theory

• Steven Costenoble, Stefan Waner, Equivariant ordinary homology and cohomology, Springer, 2016 (arXiv:math/0310237)

• Germán Stefanich: Chern Character in Twisted and Equivariant K-Theory [pdf, pdf]

As Bredon-equivariant cohomology theory

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

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

• Igor Kriz, Yunze Lu, On the $RO(G)$-graded coefficients of dihedral equivariant cohomology, Mathematical Research Letters 27 4 (2020) (arXiv:2005.01225, doi:10.4310/MRL.2020.v27.n4.a7)

for equivariance group the quaternion group:

• 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)