2021/03/07 by Nguyễn Khắc Tín, Tin, Nguyen Khac
Mathematics · #55S05 #55S10 #55T15 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2103.04393
openalex publication_date 2021/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Pk=H*((ℝP∞)k) be the modulo-2 cohomology algebra of the direct product of k copies of infinite dimensional real projective spaces ℝP∞. Then, Pk is isomorphic to the graded polynomial algebra \mathbbF2[x1,…,xk] of k variables, in which each xj is of degree 1, and let GLk be the general linear group over the prime field \mathbbF2 which acts naturally on Pk. Here the cohomology is taken with coefficients in the prime field \mathbb F2 of two elements. We study the \it hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra, A. In this Note, we explicitly compute the hit problem for k = 5 and the degree 5(2s-1)+24.2s with s an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-2 Steenrod algebra, TorAk, k+n(\mathbbF2, \mathbbF2), to the subspace of \mathbbF2⊗APk consisting of all the GLk-invariant classes of degree n. We show that Singer's conjecture for the algebraic transfer is true in the case k=5 and the above degrees. This method is different from that of Singer in studying the image of the algebraic transfer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees in the case k=6.