On a conjecture of Mahowald on the cohomology of finite sub-Hopf algebras of the Steenrod algebra (Q2183476)

As the title indicates, the author proves a conjecture of Mahowald relevant to the chromatic phenomena in the classical Adams spectral sequence. The conjecture is: \[ \mathbb F_2[v_0,v_1^{2^{2n+\varepsilon}},v_2^{2^{2n-1+\varepsilon}},\dots,v_n^{2^{n+1+\varepsilon}}, v_{n+1}^{2^{n+2}},v_{n+2}^{2^{n+3}},\dots, v_{2n+\varepsilon}^{2^{2n+1+\varepsilon}}]\subset\mathrm{Ext}_{A(2n+\varepsilon)}(\mathbb F_2, \mathbb F_2) \] for \(\varepsilon=0,1\). Here, \(A(n)\) denotes the Hopf subalgebra of the Steenrod algebra \(A\) at the prime two generated by \(Sq^0, Sq^1, Sq^2, \dots, Sq^{2^n}\), and the \(v_k\)'s correspond to the polynomial generators of the homotopy groups of the Brown-Peterson spectrum. The first attempt to the conjecture was found in the paper of \textit{M. Mahowald} and the author [Trans. Am. Math. Soc. 300, 191--206 (1987; Zbl 0625.55016)]. They showed that \({\mathbb F_2}[v_0,v_1^{4N_1},\dots, v_i^{2^{i+1}N_i},\dots, v_n^{2^{n+1}N_n}]\subset\mathrm{Ext}_{A(n)}(\mathbb F_2, \mathbb F_2)\) with \(N_n=1\). For an odd prime version, the author showed in [ibid. 309, No. 1, 77--86 (1988; Zbl 0662.55006)] that \({\mathbb F_p}[v_0,v_1^{p^n},\dots, v_i^{p^{n-i+1}},\dots, v_n^{p}]\subset\mathrm{Ext}_{A(n)}(\mathbb F_p, \mathbb F_p)\) for the Hopf subalgebra \(A(n)\) of the Steenrod algebra generated by \(\beta\), \(P^1\), \(\dots\), \(P^{p^{n-1}}\), by use of the Cartan-Eilenberg spectral sequence. The author shows the conjecture by use of the Davis-Mahowald spectral sequence rather than the Cartan-Eilenberg spectral sequence.