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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 10th 2018

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 20th 2019
• (edited Feb 20th 2019)

added 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-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 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.)

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

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.