Invariant elements in the dual Steenrod algebra
الموضوعات :
1 - Department of Mathematics, Faculty of Science, Ege University, 35100, Izmir, Turkey
2 - Department of Mathematics, Faculty of Science, Ege University, 35100, Izmir, Turkey
الکلمات المفتاحية: Steenrod algebra, cohomology operations, Hopf algebra,
ملخص المقالة :
In this paper, we investigate the invariant elements of the dual mod $p$ Steenrod subalgebra ${\mathcal{A}_p}^*$ under the conjugation map $\chi$ and give bounds on the dimensions of $(\chi-1)({\mathcal{A}_p}^*)_d$, where $({\mathcal{A}_p}^*)_d$ is the dimension of ${\mathcal{A}_p}^*$ in degree $d$.
[1] J. Adem, The iteration of Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. USA. 38 (1952), 720-726.
[2] J. Adem, The relations in Steenrod powers of cohomology classes, Algebraic geometry and topology, Symposium in honour of S. Lefschetz, Princeton University Press, 1957.
[3] H. Cartan, Sur les groupes d’Eilenberg-Mac Lane, Proc. Nat. Acad. Sci. USA. 40 (1954), 704-707.
[4] M. D. Crossley, S. Whitehouse, On conjugation invariants in the dual Steenrod algebra, Proc. Amer. Math. Soc. 128 (2000), 2809-28018.
[5] D. M. Davis, The antiautomorphism of the Steenrod Algebra, Proc. Amer. Math. Soc. 44 (1974), 235-236.
[6] I. Karaca, I. Y. Karaca, Conjugation in the mod-p Steenrod algebra. Turk. J. Math. 24 (2000), 359-365.
[7] J. Milnor, The Steenrod algebra and it’s dual, Annals. Math. 67 (1958), 150-171.
[8] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago IL, 1994.
[9] J. P. Serre, Cohomologie module 2 des complexes d’Eilenberg-Mac-Lane, Comm. Math. Helv. 27 (1953), 198-232.
[10] J. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc. 119 (1993), 657-661.
[11] N. E. Steenrod, D. B. A. Epstein, Cohomology operations, Annals of Math Studies, 50: Princeton University Press, 1962.
[12] S. D. Straffin, Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc. 49 (1975), 253-255.
[13] T. Vergili, I. Karaca, A note on the new basis in the mod 2 Steenrod algebra, J. Linear. Topological. Algebra. 7 (2) (2018), 101-107.
[14] R. M. W. Wood, Problems in the steenrod algebra, Bull. London. Math. Soc. 30 (1998), 449-517.
