A generalization of the Atiyah-Segal completion theorem
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.
scientific article; zbMATH DE number 4073960