On a minimal set of generators for the algebra $H^*(BE_6; \mathbb F_2)$ as a module over the Steenrod algebra and applications

Abstract

Let $\mathcal P_n \cong H^{*}\big(BE_n; \mathbb F_2 \big)$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This problem remains unsolvable for $n>4,$ even with the aid of computers in the case of $n=5.$ By considering $\mathbb F_2$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathbb F_2$-graded vector space $\mathbb F_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ This paper aims to explicitly determine an admissible monomial basis of the $ \mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ in the generic degree $n(2^{r}-1)+2\cdot 2^{r},$ where $r$ is an arbitrary non-negative integer, and in the case of $n=6.$ As an application of these results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degrees $(n-1)\cdot(2^{n+u-1}-1)+\ell\cdot2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$ Moreover, for any integer $r>1,$ the behavior of the sixth Singer algebraic transfer in degree $6(2^{r}-1)+2\cdot2^r$ is also discussed at the end of this paper. Here, the Singer algebraic transfer is a homomorphism from the homology of the Steenrod algebra to the subspace of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n(\mathbb F_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+*}(\mathbb F_2,\mathbb F_2).$

Keywords

• Steenrod algebra
• polynomial algebra
• hit problem
• graded rings.}
• Steenrod algebra, polynomial algebra, hit problem, graded rings

References

1. [1] J. M. Boardman, Modular representations on the homology of power of real projective space, in: M. C. Tangora (Ed.), Algebraic Topology, Oaxtepec, 1991, in: Contemp. Math. 146, 49-70, 1993.
2. [2] R.R. Bruner, L.M. Ha, , N.H.V. Hung, On behavior of the algebraic transfer, Trans. Amer. Math. Soc. 357, 473-487, 2005.
3. [3] T.W. Chen, Determination of $\text{ Ext}_{\mathcal A}^{5,*}(\mathbb Z/2,\mathbb Z/2)$, Topology Appl. 158, 660-689, 2011.
4. [4] M.D. Crossley, N.D. Turgay, Conjugation invariants in the mod 2 dual Leibniz-Hopf algebra, Comm. Algebra 41 (9), 3261-3266, 2013.
5. [5] M.D. Crossley, The Steenrod algebra and other copolynomial Hopf algebras, Bull. London Math. Soc. 32 (5), 609-614, 2000.
6. [6] A.S. Janfada, R.M.W. Wood, Generating $H^*({\rm BO}(3),\Bbb F_2)$ as a module over the Steenrod algebra, Math. Proc. Camb. Phil. Soc. 134, 239-258, 2003.
7. [7] M. Kameko, Products of projective spaces as Steenrod modules, Ph.D. Thesis, The Johns Hopkins University, ProQuest LLC, Ann Arbor, MI, 1990. 29 pp.
8. [8] W.H. Lin, $\text{ Ext}_{\mathcal A}^{4,*}(\mathbb Z/2,\mathbb Z/2)$ and $\text{ Ext}_{\mathcal A}^{5,*}(\mathbb Z/2,\mathbb Z/2)$, Topology Appl. 155, 459-496, 2008.

9. [9] J. Milnor, The Steenrod algebra and its dual, Ann. of Math. 67, 150-171, 1958.
10. [10] N. Minami, The iterated transfer analogue of the new doomsday conjecture, Trans. Amer. Math. Soc. 351, 2325-2351, 1999.
11. [11] L.X. Mong, N. Sum, The hit problem for the polynomial algebra of five variables in a type of generic degrees, Preprint, 22 pages, 2022.
12. [12] 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, 92-100, 2016.
13. [13] T.N. Nam, $\mathcal A$-générateurs génériques pour l’algébre polynomiale, Adv. Math. 186, 334-362, 2004.
14. [14] F.P. Peterson, Generators of $H^*(\mathbb RP^\infty \times \mathbb RP^\infty)$ as a module over the Steenrod algebra, Abstracts Amer. Math. Soc. 833, 55-89, 1987.
15. [15] D.V. Phuc, On Peterson’s open problem and representations of the general linear groups, J. Korean Math. Soc. 58(3), 643-702, 2021.
16. [16] S. Priddy, On characterizing summands in the classifying space of a group, I. Amer. Jour. Math. 112, 737-748, 1990.
17. [17] J. Repka, P. Selick, On the subalgebra of $H_*( (\mathbb RP^{\infty})^n; \mathbb F_2)$ annihilated by Steenrod operations, J. Pure Appl. Algebra 127, 273-288, 1998.
18. [18] J.P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg-Maclane, Comment. Math. Helv. 27, 198-232, 1953.
19. [19] W.M. Singer, The transfer in homological algebra, Math. Zeit. 202, 493-523, 1989.
20. [20] J.H. Silverman, Hit polynomials and the canonical antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc. 123, 627-637, 1995.
21. [21] N.E. Steenrod, D.B.A. Epstein, Cohomology operations, Annals of Mathematics Studies 50, Princeton University Press, Princeton N.J, 1962.
22. [22] N. Sum, The negative answer to Kameko’s conjecture on the hit problem, Adv. Math. 225, 2365-2390, 2010.
23. [23] N. Sum, N.K. Tin, Some results on the fifth Singer transfer East-West J. Math. 17(1), 70-84, 2015.
24. [24] N.Sum, On the Peterson hit problem, Adv. Math. 274, 432-489, 2015.
25. [25] N. Sum, N.K. Tin, The hit problem for the polynomial algebra in some weight vectors, Topology Appl. 290, 107579, 2021.
26. [26] N.D. Turgay, An alternative approach to the Adem relations in the mod 2 Steenrod algebra, Turkish J. Math. 38(5), 924-934, 2014.
27. [27] N.D. Turgay, S. Kaji, The ${\rm mod}\,2$ dual Steenrod algebra as a subalgebra of the ${\rm mod}\,2$ dual Leibniz-Hopf algebra, J. Homotopy Relat. Struct. 12 (3), 727-739, 2017.
28. [28] N.K. Tin, N. Sum, Kameko’s homomorphism and the algebraic transfer, C. R. Acad. Sci. Paris, Ser. I, 354, 940-943, 2016.
29. [29] N.K. Tin, Hit problem for the polynomial algebra as a module over Steenrod algebra in some degrees, Asian-European J. Math. 15 (1), 2250007, 2022.
30. [30] N.K. Tin, A note on the $\mathcal{A}$-generators of the polynomial algebra of six variables and applications, Turkish J. Math. 46(5), 1911-1926, 2022.
31. [31] N.K. Tin, On the hit problem for the Steenrod algebra in the generic degree and its applications, Rev. R. Acad. de Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 116 (2), 2022.
32. [32] V.H. Viet, Application of SAGE Computer Algebra System for the hit problem, Master thesis, Quynhon University, 2015.
33. [33] G. Walker, R.M.W. Wood, Young tableaux and the Steenrod algebra, Proceedings of the International School and Conference in Algebraic Topology Hanoi 2004, Geom. Topol. Monogr., Geom. Topol. Publ., Coventry 11, 379-397, 2007.
34. [34] R.M.W. Wood, Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Phil. Soc. 105, 307-309, 1989.
35. [35] MAGMA Computational Algebra System (version 2. 25-8), The Computational Algebra Group at the University of Sydney, 2020, http://magma.maths.usyd.edu.au/magma.
36. [36] SAGE Mathematics Software (version 5.4.1. Stein, William et al.), The Sage Development, 2012, http://www.sagemath.org.