A generalization of the Atiyah-Segal completion theorem
type: Item
instance of: scholarly article
title: A generalization of the Atiyah-Segal completion theorem
zbMATH Open document ID: 0657.55007
DOI: 10.1016/0040-9383(88)90002-X
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 compact Lie group and H a (closed) subgroup of G, let \(I^ G_ H\) be the kernel of the restriction homomorphism \(r^ G_ H: R(G)\to R(H)\). If \({\mathcal J}\) is a set of subgroups of G closed under subconjugacy, let \((K^*_ G)^{\wedge}_{{\mathcal J}}\) denote the progroup valued G-cohomology theory defined by \(K^ n_ G(X)^{\wedge}_{{\mathcal J}}=\{K^ n_ G(X_{\alpha})/JK^ n_ G(X_{\alpha})\},\) where \(X_{\alpha}\) runs over the finite subcomplexes of X and where J runs over the finite products of ideals \(I^ G_ H\) with \(H\in {\mathcal J}\). The authors prove the following generalization of the Atiyah-Segal completion theorem: If the G-map f: \(X\to Y\) restricts to a homotopy equivalence \(f^ H: X^ H\to Y^ H\) of fixed-point-sets for each \(H\in {\mathcal J}\), then \((f^*)^{\wedge}_{{\mathcal J}}: K^*_ G(Y)^{\wedge}_{{\mathcal J}}\to K^*_ G(X)^{\wedge}_{{\mathcal J}}\) is an isomorphism. This result can be used to prove the following mixed localization and completion theorem: Let \(S\subset R(G)\) be a multiplicative set, let \(I\subset R(G)\) 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 R(G). If the G-map f: \(X\to Y\) restricts to a homotopy equivalence \(f^ H: X^ H\to Y^ H\) for all \(H\in {\mathcal H}\), then \(S^{-1}(f^*)_ I^{\wedge}: S^{-1}K^*_ G(Y)_ I^{\wedge}\to S^{-1}K^*_ G(X)_ I^{\wedge}\) is an isomorphism.
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 4073960
zbMATH Keywords: representation ring
zbMATH Keywords: equivariant K-theory
zbMATH Keywords: compact Lie group
zbMATH Keywords: Atiyah- Segal completion theorem
zbMATH Keywords: fixed-point-sets
zbMATH Keywords: mixed localization and completion
reviewed by: Rolf Kultze
MaRDI profile type: Publication
full work available at URL: 0040-9383(88)90002-x
OpenAlex ID: W2001738268
Recommended article: Algebraic analogue of the Atiyah completion theorem
Recommended article: The Atiyah-Segal completion theorem for \(C^*\)-algebras
Recommended article: A generalization of the Segal conjecture
Recommended article: A FURTHER GENERALIZATION OF THE SEGAL CONJECTURE
Recommended article: A formula for \(p\)-completion by way of the Segal conjecture
Recommended article: On a generalization of a theorem of B. Segre
Recommended article: A Generalization of a Theorem of Besicovitch
Recommended article: A generalization of Szegö's theorem
Recommended article: A compactness theorem for complete separators
Recommended article: The Atiyah-Segal completion theorem in twisted \(K\)-theory
description: scientific article; zbMATH DE number 4073960