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The rank-five Peterson hit problem, the Singer transfer, and canonical Milnor-operation layers
Algebraic Topology · Submitted 13 Nov 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $t\ge0$.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\mathbb A$. | background 1.00 | 0.10 | 0.01 | asserted 0.83 |
| s1 | We study the Peterson quotient $QP_n=\mathbb F_2\otimes_{\mathbb A}P_n$ and its relation to Singer's algebraic transfer. | contribution 0.90 | 0.68 | 0.03 | asserted 0.60 |
| s2 | 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 $n\ge2$. | result 0.96 | 0.96 | 0.10 | asserted 1.00 |
| s3 | Under the monomial--divided-power pairing, the transpose adjoint yields a necessary $P_s^0$-orthogonality condition for functionals on the non-hit quotient. | result 0.48 | 0.76 | 0.02 | asserted 0.96 |
| s4 | 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. | contribution 0.60 | 0.94 | 0.04 | asserted 1.00 |
| s5 | We apply these constructions to the rank-five degree family \[ d_t=2^{t+5}+2^{t+2}+2^{t+1}-5 \qquad (t\ge0). | contribution 0.91 | 0.70 | 0.03 | asserted 0.76 |
| s6 | \] 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. | result 1.00 | 0.89 | 0.10 | asserted 1.00 |
| s7 | 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 $t\ge0$. | result 0.89 | 0.83 | 0.80 | asserted 1.00 |
| s8 | 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. | background 0.97 | 0.13 | 0.02 | asserted 0.99 |
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Original 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.
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By using Theorem [ref], we obtain an isomorphism that can be used to compute the desired this invariant space: [equation] This shows that [equation] Thus, to obtain the dimension of $((QP_5)_33)^GL_5,$ we need to determine the $GL_5$-invariants $((QP_5)_33(ω_(i)))^GL_5$ for $i =1, 3$ as well as $((QP_5)_14)^GL_5.$ The description of $((QP_5)_14)^GL_5$ can be found in our previous work [ref] , where the following theorem is stated.
2412.02494 · Key outcomes and their respective demonstrations · 0.80
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Also, for a more comprehensive understanding of our algorithmic approach, the reader may also refer to our implementation in [ref] .
Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory · Computation of the invariant space $((QP_5)_33(ω_(3)))^GL_5$ · 0.87
Cites as background 1 paper
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A homogeneous element $f∈(P_n)_d$ is hit if there is a finite sum $f = ∑_j > 0Sq^j(f_j),$ where the homogeneous elements $f_j$ belong to $(P_n)_d-j, d> j.$ The interested reader may consult the following references for further information on hit problems: Crabb-Hubbuck [ref] , Kameko [ref] , Monks [ref] , Moetele-Mothebe [ref] , Phúc and Sum [ref] , Phúc [ref] , Sum [ref] , Repka-Selick [ref] , Singer [ref] , Walker-Wood [ref] , Wood [ref] , and others.
2412.02494 · Introduction · 1.00
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