A note on the new basis in the mod 2 Steenrod algebra
الموضوعات :
1 - Department of Mathematics, Faculty of Science, Ege University, Turkey
2 - Department of Mathematics, Faculty of Science, Ege University, Turkey
الکلمات المفتاحية: Steenrod algebra, cohomology operations, Hopf algebra,
ملخص المقالة :
The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations, denoted by $Sq^n$, between the cohomology groups with $\mathbb{Z}_2$ coefficients of any topological space. Regarding to its vector space structure over $\mathbb{Z}_2$, it has many base systems and some of the base systems can also be restricted to its sub algebras. On the contrary, in addition to the work of Wood, in this paper we define a new base system for the Hopf subalgebras $\mathcal{A}(n)$ of the mod $2$ Steenrod algebra which can be extended to the entire algebra. The new base system is obtained by defining a new linear ordering on the pairs $(s+t,s)$ of exponents of the atomic squares $Sq^{2^s(2^t-1)}$ for the integers $s\geq 0$ and $t\geq 1$.
[1] D. Arnon, Monomial bases in the Steenrod algebra, J. Pure. Appl. Algebra. 96 (1994), 215-223.
[2] I. Karaca, Monomial bases in the mod p Steenrod algebra, Czechoslovak Math. 55 (2005), 699-707.
[3] H. Margolis, Spectra and the Steenrod algebra, North Holland Math Library vol.29 Elsevier Amsterdam, 1983.
[4] J. Milnor, The Steenrod algebra and it's dual, Annals of Math. 67 (1958), 150-171.
[5] K. G. Monks, Change of basis, monomial relations, and the P^s_t bases for the Steenrod algebra, J. Pure. Appl.
Algebra. 125 (1998), 235-260.
[6] J. H. Palmieri, J. J. Zhang, Commutators in the Steenrod algebra, New York J. Math. 19 (2013), 23-27.
[7] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan's xed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago IL, 1994.
[8] N. E. Steenrod, D. B. A. Epstein, Cohomology operations, Annals of Math Studies 50 Princeton University Press, 1962.
[9] R. M. W. Wood, Problems in the steenrod algebra, Bull. London. Math. Soc. 30 (1998), 449-517.
[10] R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. London. Math. Soc. 27 (1995), 380-386.
