Journal of Homotopy and Related Structures

, Volume 11, Issue 3, pp 469–491 | Cite as

The mod 2 Hopf ring for connective Morava K-theory

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Abstract

This paper examines the mod 2 homology of the spaces in the Omega-spectrum for connective Morava K-theory, i.e., the mod 2 Hopf ring for connective Morava K-theory. A natural set of generators for this Hopf ring arising from the homology and homotopy of the connective Morava K-theory spectrum is calculated and the non-trivial circle product relations among the generators arising from homology and homotopy are determined. This Hopf ring calculation is accomplished using Dieudonné ring theory and Adams spectral sequences for the connective Morava K-theory of Brown–Gitler spectra.

Keywords

Hopf ring Dieudonné ring Morava K-theory  Brown–Gitler spectra 

Mathematics Subject Classification

Primary 55T25 55P42 55P47 55S12 55S15 

References

  1. 1.
    Adams, J.F.: Stable homotopy and generalised homology. In: Proceedings of Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, Reprint of the 1974 original (1995)Google Scholar
  2. 2.
    Bousfield, A.K., Curtis, E.B., Kan, D.M., Quillen, D.G., Rector, D.L., Schlesinger, J.W.: The mod-p lower central series and the Adams spectral sequence. Topology 5, 331–342 (1966)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brown Jr, E.H., Gitler, S.: A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Topology 12, 283–295 (1973)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brown, E.H., Peterson, F.P.: On the stable decomposition of \(\Omega ^{2} S^{3}\). Trans. Am. Math. Soc. 243, 287–298 (1978)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: H\(_{\infty }\)ring spectra and their applications. In: Proceedings of Lecture Notes in Mathematics, vol. 1176. Springer, Berlin (1986)Google Scholar
  6. 6.
    Buchstaber, V., Lazarev, A.: Dieudonné modules and p-divisible groups associated with Morava K-theory of Eilenberg–Mac Lane spaces. Algebr. Geom. Topol. 7, 529–564 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cohen, R.L.: Odd primary infinite families in stable homotopy theory, Mem. Am. Math. Soc. 30, viii+92 (1981)Google Scholar
  8. 8.
    Goerss, P.G., Lannes, J., Morel, F.: Hopf algebras, Witt vectors, and Brown–Gitler spectra. In: Proceedings of Algebraic Topology (Oaxtepec, 1991), Contemporary Mathematics, vol. 146, pp. 111–128. American Mathematical Society, Providence (1993)Google Scholar
  9. 9.
    Goerss, P.G.: Hopf rings, Dieudonné modules, and \(E_*\Omega ^2S^3\) in Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemporary Mathematics, vol. 239, pp. 115–174. American Mathematical Society, Providence (1999)Google Scholar
  10. 10.
    Hara, S.I.: The Hopf rings for connective Morava K-theory and connective complex \(K\)-theory. J. Math. Kyoto Univ. 31, 43–70 (1991)MathSciNetMATHGoogle Scholar
  11. 11.
    Hunton, J.R., Turner, P.R.: Coalgebraic algebra. J. Pure Appl. Algebra 129, 297–313 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kitchloo, N., Laures, G., Wilson, W.S.: The Morava \(K\)-theory of spaces related to BO. Adv. Math. 189, 192–236 (2004)Google Scholar
  13. 13.
    Mahowald, M.E.: A new infinite family in \({}_{2}\pi _{*}{}^s\). Topology 16, 249–256 (1977)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Margolis, H.R.: Spectra and the Steenrod Algebra, North-Holland Mathematical Library, vol. 29. North-Holland, Amsterdam (1983)Google Scholar
  15. 15.
    Miller, H.R.: The Sullivan conjecture on maps from classifying spaces. Ann. Math (2) 120, 39–87 (1984)Google Scholar
  16. 16.
    Milnor, J.: The Steenrod algebra and its dual. Ann Math. (2) 67, 150–171 (1958)Google Scholar
  17. 17.
    Pearson, P.T.: The connective real K-theory of Brown–Gitler spectra. Algebr. Geom. Topol. 14, 597–625 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ravenel, D.C.: Complex Cobordism and Stable Homotopy Groups of Spheres, AMS Chelsea Series, vol. 347. AMS Chelsea Publishing, Providence (2003)CrossRefGoogle Scholar
  19. 19.
    Ravenel, D.C.: The homology and Morava \(K\)-theory of \(\Omega ^2 SU(n)\). Forum Math. 5, 1–21 (1993)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Ravenel, D.C., Wilson, W.S.: The Hopf ring for complex cobordism. J. Pure Appl. Algebra 9(1976/77), 241–280Google Scholar
  21. 21.
    Schoeller, C.: Étude de la catégorie des algèbres de Hopf commutatives connexes sur un corps. Manuscripta Math. 3, 133–155 (1970)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Strickland, N.P.: Bott periodicity and Hopf rings, Ph.D. Thesis, University of Manchester (1993). http://neil-strickland.staff.shef.ac.uk/papers/thesis.dvi
  23. 23.
    Wilson, S.W.: The Hopf ring for Morava K-theory. Publ. Res. Inst. Math. Sci. 20, 1025–1036 (1984)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wilson, W.S.: Hopf rings in algebraic topology. Expo. Math. 18, 369–388 (2000)MathSciNetMATHGoogle Scholar
  25. 25.
    Yamaguchi, A.: Morava K-theory of double loop spaces of spheres. Mathematische Zeitschrift 199, 511–523 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Tbilisi Centre for Mathematical Sciences 2015

Authors and Affiliations

  1. 1.Department of MathematicsHope CollegeHollandUSA

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