nLab Burnside ring is equivariant stable cohomotopy of the point

Context

Representation theory

representation theory

geometric representation theory

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 $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation 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
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

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.