# nLab Atiyah-Segal completion theorem

Contents

cohomology

### Theorems

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

By the general discussion at equivariant K-theory, given a suitable topological group $G$ with an action on a topological space $X$ there is a canonical map

$K_G(X) \to K_G^{Bor}(X) \simeq K_G(X \times E G) \simeq K(X \!\sslash\! G)$

from the equivariant K-theory of $X$ to the ordinary topological K-theory of the homotopy quotient (Borel construction).

While this map is never an isomorphism unless $G$ is the trivial group, the Atiyah-Segal completion theorem says that this map exhibits $K(X//G)$ as the formal completion of the ring $K_G(X)$ at the augmentation ideal of the representation ring of $G$ (hence, regarded as a ring of functions, the restriction to an infinitesimal neighbourhood of the base point).

The analog stable for stable cohomotopy is the Segal-Carlsson completion theorem:

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

## Consequences

In the case where $X=*$ (i.e. a point), we have that $K_G(*) \simeq R(G)$ and $K(*//G)=K(BG)$, thus we conclude that

$K(BG) \simeq R(G)\hat{_I}$

where $I$ is the augmentation ideal of the representation ring of $G$.

## References

The original articles are

• Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam)

• Michael Atiyah, Friedrich Hirzebruch, Vector bundle and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 (pdf)

• Michael Atiyah, Graeme Segal, Equivariant $K$-theory and completion, J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. (Euclid)

Reviews and surveys include

Last revised on July 8, 2019 at 18:37:28. See the history of this page for a list of all contributions to it.