# 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 $K(*//G)=KU^0(B G)$, thus we conclude that

$KU^0(BG) \simeq R(G)\hat{_I}$

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

## Examples and applications

###### Example

(complex topological K-theory of $B S^1$)

The complex topological K-theory of the classifying space $B S^1$ of the circle group is the power series ring:

$KU^0 \big( B S^1 \big) \;\simeq\; \mathbb{Z} [ [ c_1^{KU} ] ] \,,$

where $c_1^{K U}$ is any complex orientation of KU (see this Prop).

On the other hand, the representation ring of the circle group is (see this Example)

$R_{\mathbb{C}}(S^1) \;\simeq\; \mathbb{Z}[x, x^{-1}] \,.$

Here $x$ is the class of the 1-dimensional irrep $U(1) \hookrightarrow \mathbb{C}$, the augmentation ideal is clearly generated by $c \coloneqq x - 1$. The corresponding completion is again

\begin{aligned} \big( R_{\mathbb{C}}(S^1) \big)\hat{_{(c)}} & \;\simeq\; \mathbb{Z}[x, x^{-1}][ [ c ] ] / (c - x + 1) \\ & \;\simeq\; \mathbb{Z}[ [ c ] ][ (1 + c)^{-1} ] \\ & \;\simeq\; \mathbb{Z}[ [ c ] ] \,, \end{aligned}

where the first step is this example and the third step observes that $1 + c$ is invertible in the $c$-power series ring (by this Prop.).

(e.g. Buchholtz 08, Sec. 8.2, also Math.SE:a/3282578)

compare also Greenlees 1994, p. 74

## References

The original articles:

Review and survey:

Last revised on June 22, 2021 at 08:20:15. See the history of this page for a list of all contributions to it.