2021/12/01 by Đặng Võ Phúc, Phuc, Dang Vo
Mathematics · Physics and Astronomy · #Homotopy and Cohomology in Algebraic Topology #Black Holes and Theoretical Physics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2203.03703
We denote by \mathbb Z2 the prime field of two elements and by Pt = \mathbb Z2[x1, …, xt] the polynomial algebra of t generators x1, …, xt with the degree of each xi being one. Let \mathcal A2 be the Steenrod algebra over \mathbb Z2. A central problem of homotopy theory is to determine a minimal set of generators for the "cohits" space \mathbb Z2⊗\mathcal A2 Pt. This problem, which is called the "hit" problem for Steenrod algebra, has been systematically studied for t≤ 4. The present paper is devoted to the investigation of the structure of \mathbb Z2⊗\mathcal A2 Pt in some certain "generic" degrees. More specifically, we explicitly determine a monomial basis of \mathbb Z2⊗\mathcal A2 P5 in degree ns=5(2s-1) + 42.2s for every non-negative integer s. As a result, it confirms Sum's conjecture [14] for a relation between the minimal sets of \mathcal A2-generators of the algebras Pt-1 and Pt in the case t=5 and degree ns. As applications, we obtain the dimension of \mathbb Z2⊗\mathcal A2 P6 in the generic degree 5(2s+5-1) + n0.2s+5 for all s≥ 0, and show that the Singer's cohomological transfer [11] is an isomorphism in bidegree (5, 5+ns).