Topology
On the hit problem for the polynomial algebra
Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 565-568.

We study the hit problem, set up by F. Peterson, of finding a minimal set of generators for the polynomial algebra Pk:=F2[x1,x2,,xk] as a module over the mod-2 Steenrod algebra, A. In this Note, we study a minimal set of generators for A-module Pk in some so-called generic degrees and apply these results to explicitly determine the hit problem for k=4.

Nous étudions le problème suivant soulevé par F. Peterson : déterminer un système minimal de générateurs comme module sur lʼalgèbre de Steenrod pour lʼalgèbre polynomiale Pk:=F2[x1,x2,,xk], problème appelé hit problem en anglais. Dans ce but, nous étudions un ensemble minimal de générateurs pour le A-module Pk dans certains degrés dits génériques. En appliquant ces résultats, nous déterminons explicitement le hit problem pour k=4.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.016
Sum, Nguyễn 1

1 Department of Mathematics, Quy Nhơn University, 170 An Dương Vương, Quy Nhơn, Bi`nh Định, Viet Nam
@article{CRMATH_2013__351_13-14_565_0,
     author = {Sum, Nguyễn},
     title = {On the hit problem for the polynomial algebra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {565--568},
     publisher = {Elsevier},
     volume = {351},
     number = {13-14},
     year = {2013},
     doi = {10.1016/j.crma.2013.07.016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.07.016/}
}
TY  - JOUR
AU  - Sum, Nguyễn
TI  - On the hit problem for the polynomial algebra
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 565
EP  - 568
VL  - 351
IS  - 13-14
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.07.016/
DO  - 10.1016/j.crma.2013.07.016
LA  - en
ID  - CRMATH_2013__351_13-14_565_0
ER  - 
%0 Journal Article
%A Sum, Nguyễn
%T On the hit problem for the polynomial algebra
%J Comptes Rendus. Mathématique
%D 2013
%P 565-568
%V 351
%N 13-14
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.07.016/
%R 10.1016/j.crma.2013.07.016
%G en
%F CRMATH_2013__351_13-14_565_0
Sum, Nguyễn. On the hit problem for the polynomial algebra. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 565-568. doi : 10.1016/j.crma.2013.07.016. http://www.numdam.org/articles/10.1016/j.crma.2013.07.016/

[1] Carlisle, D.P.; Wood, R.M.W. The boundedness conjecture for the action of the Steenrod algebra on polynomials, Manchester, 1990 (Ray, N.; Walker, G., eds.) (London Math. Soc. Lecture Notes Ser.), Volume vol. 176, Cambridge Univ. Press, Cambridge (1992), pp. 203-216 (MR1232207)

[2] Crabb, M.C.; Hubbuck, J.R. Representations of the homology of BV and the Steenrod algebra II, Sant Feliu de Guíxols, 1994 (Progr. Math.), Volume vol. 136, Birkhäuser Verlag, Basel, Switzerland (1996), pp. 143-154 (MR1397726)

[3] Kameko, M. Products of projective spaces as Steenrod modules, Johns Hopkins University, 1990 (PhD thesis)

[4] Kameko, M. Generators of the cohomology of BV4, Toyama University, 2003 (preprint)

[5] Nam, T.N. A-générateurs génériques pour lʼalgèbre polynomiale, Adv. Math., Volume 186 (2004), pp. 334-362 (MR2073910)

[6] Nam, T.N. Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 5, pp. 1785-1837 (MR2445834)

[7] Peterson, F.P. Generators of H(RP×RP) as a module over the Steenrod algebra, Abstr. Amer. Math. Soc., Volume 833 (April 1987)

[8] Priddy, S. On characterizing summands in the classifying space of a group, I, Amer. J. Math., Volume 112 (1990), pp. 737-748 (MR1073007)

[9] Repka, J.; Selick, P. On the subalgebra of H((RP)n;F2) annihilated by Steenrod operations, J. Pure Appl. Algebra, Volume 127 (1998), pp. 273-288 (MR1617199)

[10] Singer, W.M. The transfer in homological algebra, Math. Z., Volume 202 (1989), pp. 493-523 (MR1022818)

[11] Singer, W.M. On the action of the Steenrod squares on polynomial algebras, Proc. Amer. Math. Soc., Volume 111 (1991), pp. 577-583 (MR1045150)

[12] Steenrod, N.E.; Epstein, D.B.A. Cohomology Operations, Ann. of Math. Stud., vol. 50, Princeton University Press, Princeton, NJ, 1962 (MR0145525)

[13] N. Sum, The hit problem for the polynomial algebra of four variables, Quy Nhơn University, Viet Nam, 2007, preprint, 240 pages.

[14] Sum, N. The negative answer to Kamekoʼs conjecture on the hit problem, C. R. Acad. Sci. Paris, Ser. I, Volume 348 (2010), pp. 669-672 (MR2652495)

[15] Sum, N. The negative answer to Kamekoʼs conjecture on the hit problem, Adv. Math., Volume 225 (2010), pp. 2365-2390 (MR2680169)

[16] Wood, R.M.W. Steenrod squares of polynomials and the Peterson conjecture, Math. Proc. Cambridge Philos. Soc., Volume 105 (1989), pp. 307-309 (MR0974986)

Cited by Sources:

The work was supported in part by a grant of NAFOSTED.