Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof this theorem, which is not obscured by tom Diecks generalisation to compact Lie groups. Who knows a good reference? I can not even find the right paper of Segal. Where did the original proof appear?
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Here's a conceptual answer, which can be filled in to give a proof. First, go back to the non-equivariant setting: why is $\pi_0(S) = \lim \pi_N(S^N) \cong {\mathbb Z}$? because one can use transversality arguments (originally due to Pontryagin) to show that group is the same as the cobordism group of ``oriented'' zero-dimensional manifolds, which is then obviously the integers. Similarly, the cobordism of oriented zero-dimensional $G$-manifolds is isomorphic to the Burnside ring, so the result you seek follows once you can establish some transversality for maps $S^V \to S^V$. (These transversality results are notoriously complicated and not always valid in the equivariant setting, but this one doesn't present much trouble). |
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This is from Segal's paper Equivariant stable homotopy theory. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59--63. Gauthier-Villars, Paris, 1971. MR0423340 (54 #11319) where he states that the equivariant stable cohomotopy of $S$ is $\pi_G^{-\ast}(S)=\bigoplus_K\pi_{\ast}^s(B(N_G(K)/K)^+)$. In particular, $\pi_G^0(S)$ is the Burnside ring. |
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