Document Type: Research Paper

Authors

Department of Mathematics, Faculty of Science, Ege University, Turkey

Abstract

‎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 $t\geq 1$‎.

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[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.