A formula for \(p\)-completion by way of the Segal conjecture (Q2104986)

Let \(G\) be a finite group with a Sylow \(p\)-subgroup \(S\), and let \(\mathcal{F}_G\) denote the corresponding \(p\)-fusion system. Broto-Levi-Oliver showed that one can associate a clasifying spectrum with \(\mathcal{F}_G\), which is homotopy equivalent to the \(p\)-completion of the suspension spectrum of \(BG\). The solution of the Segal conjecture enabled a description of the group of stable homotopy classes of maps \([\Sigma^\infty_+BG, \Sigma^\infty_+BH]\) in terms of the completion of the Burnside module \(A(G,H)\) of \((G,H)\)-bisets that are free with respect to the \(H\)-action at the augmentation ideal \(I_G\) in the Burnside ring \(A(G)\). \textit{K. Ragnarsson} [Algebr. Geom. Topol. 6, 195--252 (2006; Zbl 1098.55012)] defined for finite groups \(G\) and \(H\) with Sylow \(p\)-subgroups \(S\) and \(T\), respectively, a Burnside module for fusion systems \(A\mathbb{F}_p(\mathcal{F}_G,\mathcal{F}_H)\) as a submodule of the module \(A(S,T)^\wedge_p\) (i.e. the \(p\)-completion of the Burnside modules \(A(S,T)\)) of all those bisets that are \textit{fusion stable}. He showed that the natural restriction map \(A(G,H)\to A(S,T)\) factors through \(A\mathbb{F}_p(\mathcal{F}_G,\mathcal{F}_H)\) and that \(A\mathbb{F}_p(\mathcal{F}_G,\mathcal{F}_H)\) is isomorphic to the group of stable maps \([\Sigma^\infty_+BG, \Sigma^\infty_+BH]\). (In Ragnarsson's work, this is done in more generally for abstract saturated fusion systems.) Using the \(p\)-completion functor one forms the composite \[ A(G,H)\to A(G,H)^\wedge_{I_G}\cong [\Sigma^\infty_+BG, \Sigma^\infty_+BH]\to[(\Sigma^\infty_+BG)^\wedge_p, (\Sigma^\infty_+BH)^\wedge_p]\cong A\mathbb{F}_p(\mathcal{F}_G,\mathcal{F}_H), \] which is denoted \(\widehat{(-)}\). The goals of this paper is to produce a purely algebraic formula for this map. Let \({}_TH_T\) denote the underlying set of \(H\) with the two sided action of \(T\). Then this \((T,T)\)-biset is the restriction to \(T\) of the biset \({}_H H_H\), and is therefore fusion stable and can be considered as an element \({}_{\mathcal{F}_H}H_{\mathcal{F}_H}\) of \(A\mathbb{F}_p(\mathcal{F}_H,\mathcal{F}_H)\), which is invertible, since \(T\) is of index prime to \(p\) in \(H\). The main theorem in the paper, Theorem 1.1, states that the map \[ \widehat{(-)}\colon A(G,H)\to A\mathbb{F}_p(\mathcal{F}_G,\mathcal{F}_H) \] is given by composition with \({}_{\mathcal{F}_H}H_{\mathcal{F}_H}^{-1}\). Namely that if \({}_GX_H\in A(G,H)\) is a biset and \({}_{\mathcal{F}_G}X_{\mathcal{F}_H}\in A(\mathcal{F}_G,\mathcal{F}_H)\) is its image, then \[ \widehat{{}_GX_H} = ({}_{\mathcal{F}_H}H_{\mathcal{F}_H})^{-1}\circ {}_{\mathcal{F}_G}X_{\mathcal{F}_H}. \] This description reproduces the \(p\)-completion map on homotopy classes of classifying spectra, under the given identifications.