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

Abstract

Let $V_n=(\mathbb F_2)^n$, let $P_n=H^*(BV_n;\mathbb F_2)=\mathbb F_2[u_1,\ldots,u_n]$, and regard $P_n$ as an unstable module over the mod $2$ Steenrod algebra $\mathbb A$. We study the Peterson quotient $QP_n=\mathbb F_2\otimes_{\mathbb A}P_n$ and its relation to Singer's algebraic transfer. For the Milnor basis element $P_s^0=\operatorname{Sq}(0,\ldots,0,1)$, 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 $n\ge2$. 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 $\dim(QP_5)_{33}=1322$, $\dim(QP_5)_{d_t}=2841$ for $t\ge1$, and one-dimensional $GL_5$-invariant spaces in all these degrees. Together with the known nonzero classes $h_{t+1}d_{t+1}$ in $\operatorname{Ext}_{\mathbb A}^{5,5+d_t}(\mathbb F_2,\mathbb F_2)$ and their detection by the total transfer, this proves that the fifth algebraic transfer is an isomorphism in degree $d_t$ for every $t\ge0$. 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.