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  "body": "\n## Statement\n\nLet $G$ be a finite group, let $BG$ be its [[classifying space|7XRBR7AV3C]], and let $BG_+$ be $BG$ with a disjoint basepoint added.\nLet $A(G)$ be the Burnside ring of $G$.\nLet $I \\subseteq A(G)$ be its augmentation ideal, the kernel of the ring map $A(G) \\to \\Z$ that sends a finite $G$-set to its size.\nLet $\\pi^0_s$ denote stable cohomotopy in degree $0$.\nThen there is a natural isomorphism\n$$\\pi^0_s(BG_+) \\cong A(G)^\\wedge_I,$$\nwhere $A(G)^\\wedge_I = \\varprojlim_k A(G)/I^k$ is the $I$-adic completion.\n\n## Remarks\n\nThe 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 $EG \\times_G S \\to BG$.\nThe theorem says that this map induces an isomorphism after $I$-adic completion.\nLin proved the case $G = \\Z/2$ and Gunawardena the case $G = \\Z/p$ for $p$ odd.\nCarlsson proved the conjecture for all finite groups in 1984, after a reduction to elementary abelian $p$-groups.\nThe 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.\n\nAt 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.\n\n## Explanations\n\n### L1 human\n\nThe Burnside ring of a finite group is built from the finite sets on which the group acts.\nStable cohomotopy is the most basic generalized cohomology theory, built from maps to spheres.\nThe Segal conjecture says that the stable cohomotopy of the classifying space of $G$ is the Burnside ring, after a completion.\nSo pure algebra about $G$-sets computes a hard homotopy-theoretic invariant.\n\n### L2 human\n\nThe 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 $I$.\nIt 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.\nIt is parallel to the [[Sullivan conjecture|9EA7KJ50AE]], which compares fixed points and homotopy fixed points for spaces instead of spectra.\n",
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