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    • 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

    v1, current

    • 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 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

    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.)

    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.

    diff, v3, current

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