nLab Burnside ring is equivariant stable cohomotopy of the point



Representation theory



Special and general types

Special notions


Extra structure





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
equivariant cohomology
of the point *\ast
of classifying space BGB G
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
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)
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)
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="">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
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="">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)



(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.


Last revised on February 20, 2019 at 14:50:03. See the history of this page for a list of all contributions to it.