Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower (Q2822140)

This paper concerns the following question: What modules over the mod \(p\) Steenrod algebra \(A\) can be realized as the reduced mod \(p\) cohomology \(H^{*}X\) of a space \(X\) ?NEWLINENEWLINEThe structure theory of the category \(\mathcal{U}\) of unstable \(A\)-modules allows the formulation of precise questions in the case where \(H^{*}X\) is nilpotent (see [\textit{N. J. Kuhn}, Ann. Math. (2) 141, No. 2, 321--347 (1995; Zbl 0849.55022)], [\textit{L. Schwartz}, Invent. Math. 134, No. 1, 211--227 (1998; Zbl 0919.55007); erratum ibid. 182, No. 2, 449--450 (2010)]).NEWLINENEWLINEThe aim of this paper is to show how the analysis of the first two non-trivial columns of the spectral sequence associated to the Arone-Goodwillie tower for the functor \(X \mapsto \Sigma^{\infty}\Omega^{n}X\) imposes conditions on the first two non-trivial layers of the nilpotent filtration of \(H^{*}X\). The case \(n = 1\) is a generalization of the main result of \textit{N. T. Cuong} et al. [Fundam. Math. 234, No. 2, 139--161 (2016; Zbl 1360.55015)]. The case \(n > 1\) is new.