A generalization of the Segal conjecture
type: Item
instance of: scholarly article
title: A generalization of the Segal conjecture
zbMATH Open document ID: 0657.55017
DOI: 10.1016/0040-9383(88)90003-1
author: J. F. Adams
author: Jean-Pierre A. Haeberly
author: Stefan Jackowski
author: J. P. May
published in: Topology
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 4073970
zbMATH Keywords: finite group
zbMATH Keywords: Burnside ring
zbMATH Keywords: Segal conjecture
zbMATH Keywords: progroup-valued equivariant cohomotopy
Wikidata QID: Q123020761
reviewed by: Rolf Kultze
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90003-1
OpenAlex ID: W2092670057
description: scientific article; zbMATH DE number 4073970