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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 10th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexam giving this statement its own page, for ease of linking from various other entries, such as _[[Burnside ring]]_, _[[equivariant stable cohomotopy]]_, _[[Segal-Carlsson completion theorem]]_ <a href="https://ncatlab.org/nlab/revision/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">current</a>

   am giving this statement its own page, for ease of linking from various other entries, such as Burnside ring, equivariant stable cohomotopy, Segal-Carlsson completion theorem

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 20th 2019
   • (edited Feb 20th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded more details on how the identification actually works, identifying [[Burnside characters]] with winding numbers on fixed loci: *** More in detail, for $G$ a [[finite group]], this [[isomorphism]] identifies the [[Burnside character]] on the left with the [[fixed locus]]-[[degree of a continuous function|degrees]] on the right, hence for all [[subgroups]] $H \subset G$ 1. the $H$-[[Burnside marks]] $\left\vert S^H \right\vert \in \mathbb{Z}$ of [[virtual representation|virtual]] [[finite set|finite]] [[G-sets]] $S$ (which, as $H \subset G$ ranges, completely characterize the [[G-set]], by [this Prop.](table+of+marks#BurnsideCharacterIsInjective)) 1. the [[degree of a continuous function|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.](Hopf+degree+theorem#EquivariantHopfDegreeTheorem)) \[ \array{ A(G) &\underoverset{\simeq}{LD}{\longrightarrow}& \underset{\longrightarrow_{\mathrlap{V}}}{\lim} \;\; \left( \pi_0 \mathrm{Maps}^{\{0\}/} \left( S^V, S^V \right)^G \right) &=& \mathbb{S}_G(\ast) \\ S &\mapsto& LD(S) \\ \underset{ \mathclap{ \text{Burnside character} } }{ \underbrace{ \left( H \mapsto \left\vert S^H \right\vert \right) } } &=& \underset{ \mathclap{ \text{degrees on fixed strata} } }{ \underbrace{ \left( H \;\mapsto\; deg \left( S^{ dim\left( V^H\right) } \overset{\big(LD(S)\big)^H}{\longrightarrow} S^{ dim\left( V^H\right) } \right) \right) } } } \] For $G$ a [[compact Lie group]] this correspondence remains intact, with the relevant conditions on subgroups $H$ ([[closed subset|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]]. <a href="https://ncatlab.org/nlab/revision/diff/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">current</a>

   added more details on how the identification actually works, identifying Burnside characters with winding numbers on fixed loci:

   ---

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

   1. the -Burnside marks of virtual finite G-sets

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

   2. the degrees at -fixed points of representative equivariant Cohomotopy cocycles

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

   For a compact Lie group this correspondence remains intact, with the relevant conditions on subgroups (closed subgroups such that the Weyl group is a finite group) and generalizing the Burnside marks to the Euler characteristic of fixed loci.

   diff, v3, current