nLab Burnside ring is equivariant stable cohomotopy of the point

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

Statement

Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

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

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

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

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

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

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

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

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

For $G$ a compact Lie group this correspondence remains intact, with the relevant conditions on subgroups $H$ (closed subgroups such that the Weyl group $W_G(H) \coloneqq N_G(H)/H$ 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

• Graeme Segal, Equivariant stable homotopy theory, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971

• Tammo tom Dieck, T. Petrie, Geometric modules over the Burnside ring, Invent. Math. 47 (1978) 273-287 (pdf)

• Tammo tom Dieck, Transformation Groups and Representation Theory, Springer 1979

• Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups (arXiv:math/0504051)

• Stefan Schwede, section 6 of Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)