The rank-five Peterson hit problem, the Singer transfer, and canonical Milnor-operation layers

Abstract

Let $V_n=({\mathbb{F}}_2{)}^n$, let $P_n=H^{\ast }(B{V}_n;{\mathbb{F}}_2)={\mathbb{F}}_2[u_1,\dots ,u_n]$, and regard $P_n$ as an unstable module over the mod $2$ Steenrod algebra $\mathbb{A}$. We study the Peterson quotient $Q{P}_n={\mathbb{F}}_2{\otimes }_{\mathbb{A}}{P}_n$ and its relation to Singer's algebraic transfer. For the Milnor basis element $P_s^0=\mathrm{Sq}(0,\dots ,0,1)$, we prove that the kernel of the homogeneous action map $T_{s,e}:(P_n{)}_e\to (P_n{)}_{e+2^s-1}$ is exactly the Frobenius-square subspace for $0\le e\le 2^s$, and that this range is optimal when $n\ge 2$. Under the monomial--divided-power pairing, the transpose adjoint yields a necessary $P_s^0$-orthogonality condition for functionals on the non-hit quotient. We also formulate an exact dual primitive-kernel algorithm in terms of degree-lowering Steenrod operations on the divided-power algebra and give an adjoint proof of the surjectivity of Kameko's homomorphism. We apply these constructions to the rank-five degree family \[ d_t=2^{t+5}+2^{t+2}+2^{t+1}-5 \qquad (t\ge0). \] Using the certified finite-dimensional reductions recorded in the accompanying data, we obtain $\dim (Q{P}_5{)}_{33}=1322$, $\dim (Q{P}_5{)}_{d_t}=2841$ for $t\ge 1$, and one-dimensional $G{L}_5$-invariant spaces in all these degrees. Together with the known nonzero classes $h_{t+1}{d}_{t+1}$ in ${\mathrm{Ext}}_{\mathbb{A}}^{5,5+d_t}({\mathbb{F}}_2,{\mathbb{F}}_2)$ and their detection by the total transfer, this proves that the fifth algebraic transfer is an isomorphism in degree $d_t$ for every $t\ge 0$. The present paper serves as a continuation of, and a bridge to, the previous works by Vergili, Karaca, and Dougherty, as well as our recent work.

arXiv comment36 pages. Final version: adds new theoretical results and corrects an error from the original paper [Rendiconti del Circolo Matematico di Palermo, Series 2, Vol. 73 (2024), 989-1007], now retracted by the author. Title revised accordingly