Completion of \(G\)-spectra and stable maps between classifying spaces (Q555597)

Let \(G\) be a finite group and \(X\) be a bounded below \(G\)-spectrum. \(A(G)\) denotes the Burnside ring of \(G\) and \(I(G)\) the augmentation ideal of \(A(G))\). By \(X_{I(G)}^{\widehat{\;}}\) we denote the \(I(G)\)-adic completion of \(X\). Since completions are difficult to compute, the author proposes as substitute for \(X\), a \(G\)-spectrum \(X[\mathcal F_{\pmb p}]\), having the property that the natural map \(X[\mathcal F_{\pmb p}] \longrightarrow X\) becomes a weak equivalence after completion. Here \(\mathcal F_{\pmb p}\) denotes the family of all \(p\)-subgroups of \(G\) (for all primes), while \(X[\mathcal F_{\pmb p}]\) can be considered as the subspectrum of \(X\) whose isotropy groups are of prime power order. The precise definition is \[ X[\mathcal F_{\pmb p}] = E \mathcal F_{{\pmb p} +} \wedge X. \] It turns out that \[ X[{\mathcal F_{\pmb p}}]_{I(G)}^{\widehat{\;}} \] is the homotopy colimit of diagrams \[ X[{{\mathcal F_1}]_{I(G)}^{\widehat{ \;}} \longrightarrow X[\mathcal F_p]_ {I(G)}^{\widehat{ \;}}} \] for all primes \(p\) separately. Finally the author gives advices how to calculate the completion of \(X[{{\mathcal F_p}]_ {I(G)}^{\widehat{\;}}}\) for a given prime \(p\), and proves a splitting theorem for completions of specific classifying \(G\)-spectra.