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 $G$:

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 $G$).

]]>Adjusted title

]]>added the crucial pointer to

- Gunnar Carlsson,
*Equivariant Stable Homotopy and Segal’s Burnside Ring Conjecture*, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (jstor:2006940, pdf)

and a bit more

]]>