Equivariant stable homotopy and Sullivan's conjecture (Q1174470)

A proof of the Sullivan conjecture is given [\textit{H. R. Miller}, Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)]: Theorem VI.1: \(G\) a \(p\)-group, \(X\) a finite-dimensional \(G\)-complex, then \[ ({\mathbb{F}}_ p)_ \infty(X^ G)\rightarrow F_ G(EG,({\mathbb{F}}_ p)_ \infty X) \] is a weak equivalence, where \(({\mathbb{F}}_ p)_ \infty\) denotes \(\text{mod}_ p\) completion and \(F_ G(EG,-)\) homotopy fixed point set. Equivariant stable homotopy \(\omega_ G(-)\) enters the picture by introducing \(\omega_ G\)-completion. The homotopy type is that of the \(\mathbb{Z}\)-completion (Cor. II.4), but \(\omega_ G\)-completion allows for finer cosimplicial filtrations (see II, especially Cor. II.11). The layers of the filtration can be analyzed via equivariant Snaith splitting [\textit{L. G. Lewis}, \textit{J. P. May} and \textit{M. Steinberger}, Equivariant stable homotopy theory (1986; Zbl 0611.55001)]\ giving the link (sec. III) to the assertion of the Segal conjecture, that is: \(\omega_ G(X)^ G\rightarrow F_ G(EG,\omega_ G(X))\) is an equivalence after \(p\)-adic completion.