Equivariant stable homotopy and Segal's Burnside ring conjecture
This paper is the culmination of an exciting project in homotopy theory: with it the Segal conjecture becomes Carlsson's theorem, namely that if \(G\) is a finite group, then the Segal map \[ \pi_ G^{*{\hat{\;}}}(S^ 0) \to \pi^*_ S(BG^+) \] is an isomorphism, where \(\pi_ G^{*{\hat{\;}}}(S^ 0)\) denotes \(\pi^*_ G(S^ 0)\) completed at the augmentation ideal \(I(G)\) in the Burnside ring \(A(G)\). The introduction to the paper presents a careful history of work on the problem. Section 1 presents the necessary preliminaries from stable homotopy theory. The Segal conjecture is reformulated equivariantly as conjecture I.12: the map \(\pi^*_ G(S^ 0)\to \lim_{\overset \leftarrow k}\pi^*_ G(EG^{(k)+})\) becomes an isomorphism after \(I(G)\)-adic completion. Let \(V\) be a real \(G\)-module and \(S^{\infty V}=\lim_{\overset \leftarrow k}S^{kV}\). The main theorem in the paper is: let \(EG^+\to S^ 0\to \underline{EG}\) be a cofiber sequence, and suppose that the \(p\)-group \(G\) is not elementary abelian, and that conjecture I.12 holds for all \(p\)-groups \(H\) of order strictly smaller than that of \(G\). Then there is a fixed point free representation \(V\) of \(G\) so that \[ (a)\quad \{S^{\infty V},EG^+\}_*^{G,{\hat{\;}}}=0,\quad (b)\quad \{S^{\infty V},\underline{EG}\}_*^{G,{\hat{\;}}}=0. \] Then using the result of J. F. Adams, J. H. C. Gunawardena, and H. R. Miller that the Segal conjecture is true for an elementary abelian \(p\)-group \(G\), the Segal conjecture follows for a general \(p\)-group. But G. Lewis, J. P. May, and J. E. McClure have shown that this implies the Segal conjecture for a general finite group \(G\). This completes the proof of the Segal conjecture: quite a road was travelled from the early proof by W. H. Lin that the conjecture holds for \(G={\mathbb{Z}}/2{\mathbb{Z}}\). The result of D. Ravenel that the Segal conjecture holds for cyclic groups is not needed for Carlsson's proof, but Ravenel's method has independent interest.
equivariant stable cohomotopy group of spheres
scientific article; zbMATH DE number 3940263