# nLab equivariant complex cobordism cohomology theory

Contents

### Context

#### Cobordism theory

Concepts of cobordism theory

# Contents

## Idea

(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
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## Properties

### Universal orientation

For an abelian compact Lie group $G$, equivariant complex cobordism theory $MU_G$ is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 13).

Much as in the non-equivariant case (see at universal complex orientation on MU), $MU_G$ is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory $E_G$ and homotopy ring homomorphisms of $G$-spectra $MU_G \to E_G$ (Cole-Greenlees-Kriz 02, Theorem 1.2).

For the analogous statement on the equivariant Lazard ring see Greenlees 01a, Greenlees 01, Theorem 13.1, Cole-Greenlees-Kriz 02, Theorem 1.3.

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\,M B$ (B-bordism):

relative bordism theories:

algebraic:

## References

Last revised on November 20, 2020 at 06:49:23. See the history of this page for a list of all contributions to it.