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Burnside ring is equivariant stable cohomotopy of the point

Contents

Context

Representation theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

In analogy to how the representation ring of a finite group is equivalently the equivariant K-theory of the point, so the equivariant stable cohomotopy of the point is the Burnside ring.

(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)

Statement

Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

Let GG be a compact Lie group (for instance a finite group). Its Burnside ring A(G)A(G) is isomorphic to the equivariant stable cohomotopy cohomology ring 𝕊 G(*)\mathbb{S}_G(\ast) of the point in degree 0, via the Lefschwetz-Dold index:

A(G)LD𝕊 G(*). A(G) \underoverset{\simeq}{LD}{\longrightarrow} \mathbb{S}_G(\ast) \,.

More in detail, for GG a finite group, this isomorphism identifies the Burnside character on the left with the fixed locus-degrees on the right, hence for all subgroups HGH \subset G it identifies

  1. the HH-Burnside marks |S H|\left\vert S^H \right\vert \in \mathbb{Z} of virtual finite G-sets SS

    (which, as HGH \subset G ranges, completely characterize the G-set, by this Prop.)

  2. the degrees deg((LD(S)) H)deg\left( \left(LD(S)\right)^H\right) \in \mathbb{Z} at HH-fixed points of representative equivariant Cohomotopy cocycles LD(S):S VS VLD(S) \colon S^V \to S^V

    (which completely characterize the equivariant Cohomotopy-class by the equivariant Hopf degree theorem, this Prop.)

(1)A(G) LD lim V(π 0Maps {0}/(S V,S V) G) = 𝕊 G(*) S LD(S) (H|S H|)Burnside character = (Hdeg(S dim(V H)(LD(S)) HS dim(V H)))degrees on fixed strata \array{ A(G) &\underoverset{\simeq}{LD}{\longrightarrow}& \underset{\longrightarrow_{\mathrlap{V}}}{\lim} \;\; \left( \pi_0 \mathrm{Maps}^{\{0\}/} \left( S^V, S^V \right)^G \right) &=& \mathbb{S}_G(\ast) \\ S &\mapsto& LD(S) \\ \underset{ \mathclap{ \text{Burnside character} } }{ \underbrace{ \left( H \mapsto \left\vert S^H \right\vert \right) } } &=& \underset{ \mathclap{ \text{degrees on fixed strata} } }{ \underbrace{ \left( H \;\mapsto\; deg \left( S^{ dim\left( V^H\right) } \overset{\big(LD(S)\big)^H}{\longrightarrow} S^{ dim\left( V^H\right) } \right) \right) } } }

For GG a compact Lie group this correspondence remains intact, with the relevant conditions on subgroups HH (closed subgroups such that the Weyl group W G(H)N G(H)/HW_G(H) \coloneqq N_G(H)/H is a finite group) and generalizing the Burnside marks to the Euler characteristic of fixed loci.

The statement is due to Segal 71, a detailed proof making manifest the correspondence (1) is given by tom Dieck 79, theorem 8.5.1. See also tom Dieck-Petrie 78, Lück 05, theorem 1.13.

From a broader perspective of equivariant stable homotopy theory, this statement is a special case of tom Dieck splitting of equivariant suspension spectra (e.g. Schwede 15, theorem 6.14), see there.

References

Last revised on February 20, 2019 at 09:50:03. See the history of this page for a list of all contributions to it.