nLab Atiyah-Segal completion theorem

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 $K(X \!\sslash\! G)$ as the formal completion of the cohomology 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).

See also at formal completion – Examples – Atiyah-Segal theorem.

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

(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 $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

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

Related theorems

• Baum-Connes conjecture, Green-Julg theorem

• As explained in Equivariant stable homotopy theory, there is a related characterization of the K-homology of $B G$ as a localization.

References

The original articles:

• Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam:PMIHES_1961__9__23_0)

• 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:jdg/1214428815)

Review and survey:

• Ulrik Buchholtz, The Atiyah-Segal completion theorem, Master Thesis 2008 (pdf, pdf)

• Wikipedia, Atiyah-Segal completion theorem