2024/08/14 by Đặng Võ Phúc, Phuc, Dang Vo
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2408.07485
Let \mathbb S∞/\mathbb Z2 be the infinite lens space and \mathscr A be the Steenrod algebra over the binary field \mathbb F2. The cohomology H*((\mathbb S∞/\mathbb Z2)⊕ s; \mathbb F2) is known to be isomorphic to the graded polynomial ring \mathcal Ps:= \mathbb F2[x1, …, xs] on s generators of degree 1, viewed as an unstable \mathscr A-module. The Kameko squaring operation (\widetilde Sq0_*)(s; N): (\mathbb F2⊗\mathscr A \mathcal Ps)2N+s \longrightarrow (\mathbb F2⊗\mathscr A \mathcal Ps)N is rather useful in studying an open problem of determining the dimension of the indecomposables (\mathbb F2⊗\mathscr A \mathcal Ps)N. As a continuation of our recent works, this paper deals with the kernel of the Kameko (\widetilde Sq0_*)(s; Nd) for the case where s = 5 and the "generic" degree Nd is of the form Nd = 5(2d - 1) + 11.2d+1 for arbitrary d > 0. We then rectify almost all of the main results that were inaccurate in an earlier publication [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116:81 (2022)] by Nguyen Khac Tin. We have also constructed several advanced algorithms in SAGEMATH to validate our results. These new algorithms make an important contribution to tackling the intricate task of explicitly determining both the dimension and the basis for the indecomposables \mathbb F2 ⊗\mathscr A \mathcal Ps at positive degrees, a problem concerning algorithmic approaches that had not previously been addressed by any author. Also, the present study encompasses an investigation of the behavior of the cohomological transfer in bidegrees (5, 5+Nd), with the internal degree Nd mentioned above.