Equivariant stable homotopy and Segal’s Burnside ring conjecture. (English) Zbl 0586.55008

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.
Reviewer: A.Liulevicius


55Q91 Equivariant homotopy groups
55Q55 Cohomotopy groups
57S17 Finite transformation groups
55Q45 Stable homotopy of spheres
20C15 Ordinary representations and characters
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