nLab
Atiyah-Segal completion theorem

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebra

Contents

Idea

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

K G(X)K G Bor(X)K G(X×EG)K(XG) 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 XX to the ordinary topological K-theory of the homotopy quotient (Borel construction).

While this map is never an isomorphism unless GG is the trivial group, the Atiyah-Segal completion theorem says that this map exhibits K(X//G)K(X//G) as the formal completion of the ring K G(X)K_G(X) at the augmentation ideal of the representation ring of GG (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) cohomologyrepresenting
spectrum
equivariant cohomology
of the point *\ast
cohomology
of classifying space BGB G
(equivariant)
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
(equivariant)
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
(equivariant)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
(equivariant)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)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
K𝔽 1Segal 74K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)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=*X=* (i.e. a point), we have that K G(*)R(G)K_G(*) \simeq R(G) and K(*//G)=KU 0(BG)K(*//G)=KU^0(B G), thus we conclude that

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

where II is the augmentation ideal of the representation ring of GG.

Examples and applications

Example

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

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

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

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

Here xx is the class of the 1-dimensional irrep U(1)U(1) \hookrightarrow \mathbb{C}, the augmentation ideal is clearly generated by cx1c \coloneqq x - 1. The corresponding completion is again

(R (S 1))(c)^ [x,x 1][[c]]/(cx+1) [[c]][(1+c) 1] [[c]], \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+c1 + c is invertible in the cc-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.