AI 中文总结
研究五阶彼得森击中问题，通过精确稀疏消元等方法确定余击中模相关值，证明第五个辛格上同调转移是同构，找到无定向配边中49次幂的几何生成元，明确其与代数不变量的边界关系。
AI 中文摘要
彼得森击中问题旨在为多项式代数\(P_s = \mathbb{F}_2[x_1,\dots,x_s]\)作为模2斯廷罗德代数\(\mathcal{A}\)上的不稳定模寻找一组最小生成元。对于五阶情况，一般的容许基不适用，卡梅科周期性与模不变量之间的相互作用在计算上变得复杂。本文研究了一般族\(N_d = 27\cdot 2^d - 5\)中的五阶余击中模。在49次幂时通过精确稀疏消元处理292825个单项式，得到击中秩为289969，余击维数为2856。确定了精确的权重和项，证明权重为\((3,3,2,2,1)\)的和项恰好是卡梅科运算的核，维数为1891。精确不变量计算表明49次幂的一般线性群不变量形成由一个283项多项式生成的一维空间，且证明第五个辛格上同调转移在此族中是同构。从几何上看，无定向配边环的希尔伯特 - 庞加莱级数给出49次幂配边群的维数为5692。通过计算切向施蒂费尔 - 惠特尼数证明米尔诺超曲面\(H_{2,48} \subset \mathbb{R}P^2 \times \mathbb{R}P^{48}\)代表唯一的非零不可分解类，提供了一个明确的几何生成元。然而，从\(H_{2,48}\)到分类空间\(B(\mathbb{Z}/2)^5\)的明显映射将其基本类发送到具有非零\(Sq^2_*\)的同调类。因此，这个几何生成元不能与代数不变量的泛函对偶等同，在斯廷罗德理论不变线和几何配边生成元之间建立了精确的边界。
英文摘要
The Peterson hit problem seeks a minimal set of generators for the polynomial algebra $P_s = \mathbb{F}_2[x_1,\dots,x_s]$ as an unstable module over the mod-2 Steenrod algebra $\mathcal{A}$. For rank five, general admissible bases fail, and the interplay between Kameko periodicity and modular invariants becomes computationally complex. In this paper, we study the rank-five cohit module in the generic family $N_d = 27\cdot 2^d - 5$. Exact sparse elimination in degree $49$ processes $292825$ monomials, yielding a hit rank of $289969$ and a cohit dimension of $2856$. We determine the exact weight summands and prove that the weight-$(3,3,2,2,1)$ summand is exactly the kernel of Kameko's operation, with dimension $1891$. These exact values systematically correct the corresponding rank-five kernel and dimension assertions in Nguyen Khac Tin's previous paper. An exact invariant calculation shows that the general linear group invariants in degree $49$ form a one-dimensional space generated by a $283$-term polynomial, and we prove that the fifth Singer cohomological transfer is an isomorphism in this family. Geometrically, the Hilbert-Poincare series of the unoriented cobordism ring gives the dimension of the degree-$49$ cobordism group as $5692$. We prove that the Milnor hypersurface $H_{2,48} \subset \mathbb{R}P^2 \times \mathbb{R}P^{48}$ represents the unique nonzero indecomposable class by computing a tangential Stiefel-Whitney number, providing an explicit geometric generator. However, the evident map from $H_{2,48}$ to the classifying space $B(\mathbb{Z}/2)^5$ sends its fundamental class to a homology class with nonzero $Sq^2_*$. Consequently, this geometric generator cannot be identified with the functional dual of the algebraic invariant, establishing a precise boundary between the Steenrod-theoretic invariant line and the geometric cobordism generator.
Comments29 pages. This preprint has been completely restructured and expanded from arXiv:2408.07485v7, which now serves solely as a supplementary reference to the present work. Comments are welcome!
