## 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

### MSC:

 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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