Abstract
We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra \(\mathscr {A}^*\). We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of \(\mathscr {A}^*\) and its structure as a Hopf algebra.
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References
Bailey, S.M.: On the spectrum \(b{\rm o}\wedge {\rm tmf}\). J. Pure Appl. Algebra 214(4), 392–401 (2010)
Bondy, J.A., Murty, U.S.R.: Graph theory with applications. North-Holland, Elsevier Science Ltd/North-Holland (New York, Amsterdam, Oxford) (1982)
Milnor, J.: The Steenrod algebra and its dual. Ann. Math. 2(67), 150–171 (1958)
Ravenel, D.C.: Complex cobordism and stable homotopy groups of spheres. In: Pure and Applied Mathematics, vol. 121. Academic Press Inc, Orlando (1986)
Smith, L.: An algebraic introduction to the Steenrod algebra. In: Proceedings of the School and Conference in Algebraic Topology, Geom. Topol. Monogr. vol. 11, pp. 327–348. Geom. Topol. Publ., Coventry (2007)
Sweedler, M.E.: Hopf algebras. W.A. Benjamin, Inc. (New York, New York) (1969)
Walker, G., Wood, R.M.W.: Polynomials and the \({\rm mod}\, 2\) Steenrod algebra. vol. 1. The Peterson hit problem, volume 441 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2018)
Walker, G., Wood, R.M.W.: Polynomials and the \({\rm mod}\, 2\) Steenrod algebra. vol. 2. Representations of \({{\rm GL}}(n,{\mathbb{F}}_2)\), volume 442 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2018)
West, D.B.: Introduction to graph theory. Prentice Hall Inc, Upper Saddle River, NJ (1996)
Wood, R.M.W.: Differential operators and the Steenrod algebra. Proc. London Math. Soc. (3) 75(1), 194–220 (1997)
Wood, R.M.W.: Problems in the Steenrod algebra. Bull. London Math. Soc. 30(5), 449–517 (1998)
Yearwood, C.: The Dual Steenrod Algebra and Graph Theory. Senior thesis under the supervision of K. Ormsby. Reed College (2019)
Acknowledgements
The author would like to thank Caroline Yearwood, whose thesis was a major impetus for this work, and her advisor Kyle Ormsby, who kindly provided a copy of Yearwood’s thesis. The author would also like to thank Kiran Bhutani, Paul Kainen, and the anonymous referee, for their helpful comments and suggestions.
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Communicated by Anna Marie Bohmann.
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Larson, D.M. Connectedness of graphs arising from the dual Steenrod algebra. J. Homotopy Relat. Struct. 17, 145–161 (2022). https://doi.org/10.1007/s40062-022-00300-3
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DOI: https://doi.org/10.1007/s40062-022-00300-3