A generalization of the Segal conjecture
If G is a finite group, let A(G) denote the Burnside ring of G. Let \(S\subset A(G)\) be a multiplicative set, let \(I\subset A(G)\) be an ideal and define \({\mathcal H}=\cup \{Supp(P)|\) \(P\cap S=\emptyset\) and \(P\supset I\}\) where P runs over prime ideals of A(G). Using results of \textit{G. Carlsson} [Ann. Math. II. Ser. 120, 189-224 (1984; Zbl 0586.55008)] the authors prove the following generalization of the Segal conjecture: If the G-map f: \(X\to Y\) restricts to a homotopy equivalence \(f^ H: X^ H\to Y^ H\) for each \(H\in {\mathcal H}\), then \(S^{- 1}(f^*)_ I^{\wedge}: S^{-1}\pi^*_ G(Y)_ I^{\wedge}\to S^{-1}\pi^*_ G(X)_ I^{\wedge}\) is an isomorphism, where \(S^{- 1}\pi^*_ G(-)_ I^{\wedge}\) denotes the progroup-valued equivariant cohomotopy localized at S and completed at I.
scientific article; zbMATH DE number 4073970