Topological realization of level structures of the formal group law over \(\widehat{E(n)}\) (Q1585433)

Let \(\widehat{E(n)}\) be the \(I_n\)-adic complete Johnson-Wilson spectrum, and \(F_n(p)\) a spectrum whose \(\widehat{E(n)}\)-cohomology is rationally isomorphic to the extension of \(\widehat{E(n)}_*={\mathcal O}[[u_1,\dots,u_{n-1}]][u,u^{-1}]\) obtained by adjoining all roots of the \(p\)-series \([p](x)\) for the associated formal group law. Since \(F_n(p)\) contains the \(S\)-dual of the Tits building in the bottom cells, the first nontrivial mod \(p\) cohomology group of \(F_n(p)\) is the Steinberg representation. This paper is devoted to finding a homotopy equivalence \(e_nF_n(p)\simeq\Sigma^{-n}D(n)\) where \(D(n)\) denotes the cofiber of the diagonal map \(Sp^{p^{k-1}}S^0\longrightarrow Sp^{p^k}S^0\) for \(Sp^nS^0\) being the \(n\)-fold symmetric product of the sphere spectrum localized at \(p\) and \(e_n\) is the Steinberg idempotent in \({\mathbb Z}[GL_n(F_p)]\).