On a minimal set of generators for the algebra \(H^*(BE_d; F_2)\) and its applications
Volume 30, Issue 1, pp 75--88
https://doi.org/10.22436/jmcs.030.01.08
Publication Date: December 01, 2022
Submission Date: August 17, 2022
Revision Date: October 04, 2022
Accteptance Date: November 03, 2022
Authors
D. P. Phan
- Department of Mathematics, Faculty of Applied Sciences, HCMC University of Technology and Education, Viet Nam.
L. N. Hoang
- Department of Mathematics, Faculty of Applied Sciences, HCMC University of Technology and Education, Viet Nam.
T. K. Nguyen
- Department of Mathematics, Faculty of Applied Sciences, HCMC University of Technology and Education, Viet Nam.
Abstract
We investigate the Peterson hit problem for the polynomial algebra \(\mathcal P_{d}\), viewed as a graded left module over the mod-\(2\) Steenrod algebra, \(\mathcal{A}\). For \(d>4\), this problem is still unsolved, even in the case of \(d=5\) with the help of computers. In this article, we study the hit problem for the case \(d=6\) in the generic degree \(6(2^{r}-1)+6.2^r\), with \(r\) an arbitrary non-negative integer.
Furthermore, the behavior of the sixth Singer algebraic transfer in degree \(6(2^{r}-1)+6.2^r\) is also discussed at the end of this paper.
Share and Cite
ISRP Style
D. P. Phan, L. N. Hoang, T. K. Nguyen, On a minimal set of generators for the algebra \(H^*(BE_d; F_2)\) and its applications, Journal of Mathematics and Computer Science, 30 (2023), no. 1, 75--88
AMA Style
Phan D. P., Hoang L. N., Nguyen T. K., On a minimal set of generators for the algebra \(H^*(BE_d; F_2)\) and its applications. J Math Comput SCI-JM. (2023); 30(1):75--88
Chicago/Turabian Style
Phan, D. P., Hoang, L. N., Nguyen, T. K.. "On a minimal set of generators for the algebra \(H^*(BE_d; F_2)\) and its applications." Journal of Mathematics and Computer Science, 30, no. 1 (2023): 75--88
Keywords
- Polynomial algebra
- Steenrod algebra
- graded rings
MSC
References
-
[1]
J. M. Boardman, Modular representations on the homology of power of real projective space, Contemp. Math., 146 (1993), 49–70
-
[2]
A. S. Janfada, R. M. W. Wood, The hit problem for symmetric polynomials over the Steenrod algebra, Math. Proc. Cambridge Philos. Soc., 133 (2002), 295–303
-
[3]
A. S. Janfada, R. M.W.Wood, Generating H(BO(3),F2) as a module over the Steenrod algebra, Math. Proc. Cambridge Philos. Soc., 134 (2003), 239–258
-
[4]
M. Kameko, Products of projective spaces as Steenrod modules, Thesis (Ph.D.)–The Johns Hopkins University, (1990), 29 pages
-
[5]
M. F. Mothebe, P. Kaelo, O. Ramatebele, Dimension formulae for the polynomial algebra as a module over the Steenrod algebra in degrees less than or equal to 12, J. Math. Research, 8 (2016), 92–100
-
[6]
T. N. Nam, A-g´en´erateurs g´en´eriques pour l’alg`ebre polynomiale, Adv. Math., 186 (2004), 334–362
-
[7]
F. P. Peterson, Generators of H(RP1 RP1) as a module over the Steenrod algebra, Abstracts Amer. Math. Soc., 833 (1987), 55–89
-
[8]
D. V. Phuc, N. Sum, On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra, Acta Math. Vietnam., 42 (2017), 149–162
-
[9]
D. V. Phuc, A-generators for the polynomial algebra of five variables in degree 5(2t - 1) + 6.2t, Commun. Korean Math. Soc., 35 (2020), 371–399
-
[10]
D. V. Phuc, On Peterson’s open problem and representations of the general linear groups, J. Korean Math. Soc., 58 (2021), 643–702
-
[11]
S. Priddy, On characterizing summands in the classifying space of a group. I, Amer. J. Math., 112 (1990), 737–748
-
[12]
J. Repka, P. Selick, On the subalgebra of H((RP1)n; F2) annihilated by Steenrod operations, J. Pure Appl. Algebra, 127 (1998), 273–288
-
[13]
J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-MacLane, Comment. Math. Helv., 27 (1953), 198–232
-
[14]
W. M. Singer, The transfer in homological algebra, Math. Z., 202 (1989), 493–523
-
[15]
J. H. Silverman, Hit polynomials and the canonical antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc., 123 (1995), 627–637
-
[16]
N. E. Steenrod, D. B. A. Epstein, Cohomology operations, Princeton University Press, Princeton, N.J. (1962)
-
[17]
N. Sum, The negative answer to Kameko’s conjecture on the hit problem, Adv. Math., 225 (2010), 2365–2390
-
[18]
N. Sum, On the Peterson hit problem, Adv. Math., 274 (2015), 432–489
-
[19]
N. Sum, N. K. Tin, Some results on the fifth Singer transfer, East-West J. Math., 17 (2015), 70–84
-
[20]
N. Sum, N. K. Tin, The hit problem for the polynomial algebra in some weight vectors, Topology Appl., 290 (2021), 17 pages
-
[21]
N. K. Tin, The admissible monomial basis for the polynomial algebra of five variables in degree 2s+1 + 2s - 5, East-West J. Math., 16 (2014), 34–46
-
[22]
N. K. T´ın, N. Sum, Kameko’s homomorphism and the algebraic transfer, C. R. Math. Acad. Sci. Paris, 354 (2016), 940–943
-
[23]
N. K. Tin, A note on the Peterson hit problem for the Steenrod algebra, Proc. Japan Acad. Ser. A Math. Sci., 97 (2021), 25–28
-
[24]
N. K. T´ın, Hit problem for the polynomial algebra as a module over Steenrod algebra in some degrees, Asian-Eur. J. Math., 15 (2022), 21 pages
-
[25]
N. K. Tin, A note on the A-generators of the polynomial algebra of six variables and applications, Turkish J. Math., 46 (2022), 1911–1926
-
[26]
N. K. Tin, On the hit problem for the Steenrod algebra in the generic degree and its applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 116 (2022), 12 pages
-
[27]
G. Walker, R. M. W. Wood, Polynomials and the mod2 Steenrod algebra. Vol. 1. The Peterson hit problem, Cambridge University Press, Cambridge (2018)
-
[28]
G.Walker, R. M.W.Wood, Polynomials and the mod 2 Steenrod algebra. Vol. 2. Representations of GL(n, F2), Cambridge University Press, Cambridge (2018)
-
[29]
R. M. W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc., 105 (1989), 307–309
