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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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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
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 $H \subset G$
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.)
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.)
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.
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