2021/01/06 by Đặng Võ Phúc, Phuc, Dang Vo
Mathematics · #13A50 #55Q45 #55R12 #55S05 #55S10 #55T15 #Advanced Differential Equations and Dynamical Systems #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2101.11419
openalex publication_date 2021/01/06 · openalex created_date 2021/12/06 · openalex updated_date 2026/07/28
Let us consider the prime field of two elements, \mathbb F2≡ \mathbb Z2. It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra \mathscr A is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra \mathcal Pm:=\mathbb F2[x1, x2, …, xm], regarded as a connected unstable \mathscr A-module on m variables x1, …, xm, each of degree 1. The algebra \mathcal Pm is the \mathbb F2-cohomology of the product of m copies of the Eilenberg-MacLan complex K(\mathbb F2, 1). Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for m≥ 5. Our intent in this work is of studying the hit problem of five variables. More precisely, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on the hit problem for \mathscr A-module \mathcal P5 in a degree of the generic form nt:=5(2t-1) + 18.2t, for any non-negative integer t. An efficient approach to solve this problem had been presented. Two applications of this study are to determine the dimension of \mathcal P6 in the generic degree 5(2t+4-1) + n1.2t+4 for all t > 0 and to describe the modular representations of the general linear group of rank 5 over \mathbb F2. As a corollary, the cohomological "transfer", defined by William Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegree (5, 5+n0). Singer's transfer is one of the relatively efficient tools to approach the structure of mod-2 cohomology of the Steenrod algebra.