On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I
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Nguyen Khac Tin
,
Phan Phuong Dung
Abstract
Let 𝓟n := H*((ℝP∞)n) ≅ ℤ2[x1, x2, …, xn] be the graded polynomial algebra over ℤ2, where ℤ2 denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra 𝓟n, viewed as a graded left module over the mod-2 Steenrod algebra, 𝓐. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. In this article, we study the hit problem for the case n = 6 in the generic degree dr = 6(2r − 1) + 4.2r with r an arbitrary non-negative integer. By considering ℤ2 as a trivial 𝓐-module, then the hit problem is equivalent to the problem of finding a basis of ℤ2-vector space ℤ2 ⊗𝓐𝓟n. The main goal of the current article is to explicitly determine an admissible monomial basis of the ℤ2 vector space ℤ2 ⊗𝓐𝓟6 in some degrees. As an application, the behavior of the sixth Singer algebraic transfer in the degree 6(2r − 1) + 4.2r is also discussed at the end of this paper.
This research is supported by Ho Chi Minh City University of Technology and Education (HCMUTE), Vietnam
Acknowledgement
The authors would like to express our warmest thanks to Prof. Nguyen Sum (Sai Gon University, Viet Nam) for helpful conversations. The first author thanks Mr. Dang Vo Phuc (Khanh Hoa University, Viet Nam) for a helpful discussion. Finally, the authors thank the referee for helpful comments.
Communicated by Tibor Macko
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Artikel in diesem Heft
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- Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers
- A general formula in composition theory
- A nonlinear Filbert-like matrix with three free parameters: From linearity to nonlinearity
- On universality in short intervals for zeta-functions of certain cusp forms
- On asymptotics for lacunary partition functions
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- Kneser-type oscillation theorems for second-order functional differential equations with unbounded neutral coefficients
- Approximation theorems via Pp-statistical convergence on weighted spaces
- Global existence and multiplicity of positive solutions for anisotropic eigenvalue problems
- Theoretical analysis of higher-order system of difference equations with generalized balancing numbers
- On a solvable difference equations system of second order its solutions are related to a generalized Mersenne sequence
- Positive bases, cones, Helly-type theorems
- Digital Jordan surfaces arising from tetrahedral tiling
- Influence of ideals in compactifications
- On the functions ωf and Ωf
- On the 𝓐-generators of the polynomial algebra as a module over the Steenrod algebra, I
- A bivariate distribution with generalized exponential conditionals
- A note on boundary feedback stabilization for degenerate parabolic equations in multi-dimensional domains
