2026/07/21 by Dang Vo Phuc
#math.AT
The Peterson hit problem seeks a minimal set of generators for the polynomial algebra Ps = \mathbbF2[x1,…,xs] as an unstable module over the mod-2 Steenrod algebra 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 Nd = 27⋅ 2d - 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 H2,48 ⊂ ℝP2 × ℝP48 represents the unique nonzero indecomposable class by computing a tangential Stiefel-Whitney number, providing an explicit geometric generator. However, the evident map from H2,48 to the classifying space B(ℤ/2)5 sends its fundamental class to a homology class with nonzero Sq2_*. 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.