Segal conjecture

Remarks

The map comes from the natural ring map $A(G) \to \pi^0_s(BG_+)$, which sends a finite $G$-set $S$ to the transfer of the covering $EG \times_G S \to BG$. The theorem says that this map induces an isomorphism after $I$-adic completion. Lin proved the case $G = \Z/2$ and Gunawardena the case $G = \Z/p$ for $p$ odd. Carlsson proved the conjecture for all finite groups in 1984, after a reduction to elementary abelian $p$-groups. The analogous statement for complex representation rings and K-theory, $K^0(BG) \cong R(G)^\wedge_I$, is the Atiyah–Segal completion theorem, which motivated the conjecture.

At work: for the trivial group, $A(G) = \Z$ and $I = 0$, and the statement is $\pi^0_s(S^0) \cong \Z$, the degree of stable self-maps of spheres.

Going deeper

For a student

The theorem has a stronger equivariant form: the map from the fixed points of the $G$-equivariant sphere spectrum to its homotopy fixed points becomes an equivalence after completion at $I$. It gives a description of the stable maps between classifying spaces of finite groups, and it is a model for later completion and descent theorems in equivariant homotopy theory. It is parallel to the Sullivan conjecture, which compares fixed points and homotopy fixed points for spaces instead of spectra.