The gluing problem in the fusion systems of the symmetric, alternating and linear groups. (Q661862)

Let \(p\) be a prime, \(k\) be an algebraically closed field of characteristic \(p\), \(\mathcal F\) be a saturated fusion system on a \(p\)-group \(S\) and \(\mathcal F^c\) be the full subcategory of the \(\mathcal F\)-centric subgroups. We have a natural map \[ \gamma\colon H^i(\mathcal F^c;k^\times)\to\varprojlim_{[S(\mathcal F^c)]}A^i_{\mathcal F}, \] where \(A^i_{\mathcal F}\), \(i\geq 0\), is the functor which associates to an equivalence class \([P_\bullet]\) of simplices in \(\mathcal F^c\), the Abelian group \(H^i(\Aut_{\mathcal F}(P_\bullet);k^\times)\); \([S(\mathcal F^c)]\) denotes the poset of equivalence classes of simplices in \(\mathcal F^c\). We say that Linckelmann's gluing problem for an element \((\alpha_{[P_\bullet]})\) of \(\varprojlim_{[S(\mathcal F^c)]}A^i_{\mathcal F}\) has a solution if this element is in the image of \(\gamma\). The main result of this paper is that the gluing problem for any element in \(\varprojlim_{[S(\mathcal F^c)]}A^2_{\mathcal F}\) has a solution in the fusion systems of the symmetric groups and the alternating groups and also \(\text{GL}_d(p^r)\) and \(\text{SL}_d(p^r)\).