The Segal conjecture for elementary p-groups. II. p-adic completion in equivariant cohomology
The authors give a treatment, ``as efficient as present technology seems to allow'', of that stretch of the proof of the Segal conjecture which starts from the Ext calculations of the reviewer, \textit{J. H. Gunawardena} and \textit{H. Miller} [Part I, ibid. 24, 435-460 (1985; Zbl 0611.55010)] and delivers the conjecture for elementary abelian p-groups (the input to \textit{G. Carlsson}'s argument that the case of elementary abelian p-groups implies the general case [Ann. Math., II. Ser. 120, 189-224 (1984; Zbl 0586.55008)]. The central conclusion is thus that when \(G=(Z_ p)^ r\) and X is a suitable G-space, the boundary map in Carlsson's fundamental exact sequence (here written \(\delta: {\hat\pi}^ q_ G(X;\tilde EG)\to {\hat \pi}_ g^{q+1}(X;EG_+))\) is an isomorphism (Theorem D). The two groups \({\hat \pi}\) can be calculated (Theorems A, B) and are isomorphic; it remains to keep the map \(\delta\) under control. This is done by naturality; the authors obtain a commutative diagram \[ \begin{tikzcd} \hat\pi_G^{r-1} (X;\widetilde EG) \ar[r,"\delta"]\ar[d,"\eta" '] & \hat\pi_G^r (X;EG_+) \ar[d,"\eta"]\\ \hat k_G^{r-1} (X;\widetilde EG) \ar[r,"\delta" '] & \hat k_ G^r (X;EG_+) \end{tikzcd} \] in which k is a theory easier to handle, the left-hand \(\eta\) becomes a monomorphism when reduced modulo p, and the lower \(\delta\) is an isomorphism. This leads to the conclusion. In order to establish such a commutative diagram, the authors generalize most of the technology in Carlsson's proof, so that cohomotopy may be replaced by a fairly general cohomology theory. In the process, they simplify many of the techniques, giving for example a smaller model for the ``singular set''. While they are at it, it costs them little extra to prove the Segal conjecture for a general finite group G. Some of the techniques should be recommended to workers in the field.
scientific article; zbMATH DE number 4147099