nLab completion theorem for complex cobordism cohomology

Contents

Context

Algebra

higher algebra

universal algebra

Contents

Idea

As with the Atiyah-Segal completion theorem for equivariant K-theory, similar results have been developed for equivariant complex cobordism cohomology theory (Löffler 74, see Greenlees-May, theorem 1.1), and more generally for MU-modules.

For instance, for certain classes of groups, $G$, the cohomology ring $MU(B G)$ is the completion of $MU_G(\ast)$ at its augmentation ideal.

(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

The proof is due to

• Peter Löffler, Bordismengruppen unitärer Torusmannigfaltigkeiten, Manuscripta Math. 12 (1974), 307-327 (doi:10.1007/BF01171078)

• G. Comezaña, Peter May, A completion theorem in complex cobordism. In J. P. May, et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Series in Mathematics No. 91. 1996

Detailed review is in

Last revised on October 26, 2018 at 08:31:31. See the history of this page for a list of all contributions to it.