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added the crucial pointer to
and a bit more
added the actual statement to the Idea section:
The statement known as Segal’s conjecture (due to Graeme Segal in the 1970s, then proven by Carlsson 84) characterizes the stable cohomotopy groups $\pi^\bullet_{st}(B G)$ of the classifying space $B G$ of a finite group $G$ as the formal completion $\widehat \pi^\bullet_S(B G)$ at the augmentation ideal (i.e. when regarded as a ring of functions: its restriction to the infinitesimal neighbourhood of the basepoint) of the ring $\pi^\bullet_{st,G}(\ast)$ of $G$-equivariant stable cohomotopy groups of the point, the latter also being isomorphic to the Burnside ring $A(G)$ of $G$:
$A(G) \simeq \pi^\bullet_{st,G}(\ast) \overset{ \text{completion} \atop \text{projection} }{\longrightarrow} \widehat \pi^\bullet_{st,G}(\ast) \simeq \pi^\bullet_{st}(B G) \,.$This statement is the direct analogue of the Atiyah-Segal completion theorem, which makes the analogous statement for the generalized cohomology not being (equivariant) stable cohomotopy but (equivariant) complex K-theory (with the role of the Burnside ring then being the representation ring of $G$).
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