Document Type: Research Paper

Authors

Department of Mathematic‎s, ‎Faculty of Science‎, ‎Ege University‎, ‎35100‎, ‎Izmir‎, ‎Turkey

Abstract

‎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 $d$‎.

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Main Subjects

[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.