Mathematische Zeitschrift

, Volume 275, Issue 3–4, pp 953–1004 | Cite as

The homotopy of the \(K(2)\)-local Moore spectrum at the prime \(3\) revisited

  • Hans-Werner HennEmail author
  • Nasko Karamanov
  • Mark Mahowald


In this paper we use the approach introduced in (Goerss et al., Ann Math 162(2):777–822, 2005) in order to analyze the homotopy groups of \(L_{K(2)}V(0)\), the mod-\(3\) Moore spectrum \(V(0)\) localized with respect to Morava \(K\)-theory \(K(2)\). These homotopy groups have already been calculated by Shimomura (J Math Soc Japan 52(1): 65–90, 2000). The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura (J Math Soc Japan 52(1): 65–90, 2000). An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the \(K(2)\)-localization of the spectrum \(TMF\) of topological modular forms and related spectra. Even more, the Adams–Novikov differentials for \(L_{K(2)}V(0)\) can be read off from those for \(TMF\).



The authors would like to thank the Mittag-Leffler Institute, Northwestern University, Université Louis Pasteur at Strasbourg and the Ruhr-Universität Bochum for providing them with the opportunity to work together.


  1. 1.
    Behrens, M.: A modular description of the \({K}(2)\)-local sphere at the prime 3. Topology 45(2), 343–402 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Deligne, P.: Courbes elliptiques: formulaire (d’après J. Tate), Modular Fonctions of One Variable IV. Lecture Notes in Math. (Springer), vol. 476, pp. 53–73 (1975)Google Scholar
  3. 3.
    Devinatz, E.S., Hopkins, M.J.: The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts. Am. J. Math. 117(3), 669–710 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Devinatz, E.S., Hopkins, M.J.: Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43(1), 1–47 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Goerss, P., Henn, H.-W., Mahowald, M.: The homotopy of \(L_2V(1)\) for the prime 3, Categorical decomposition techniques in algebraic topology (Isle of Skye). Progr. Math. 215, 125–151 (2001)MathSciNetGoogle Scholar
  6. 6.
    Goerss, P., Henn, H.-W., Mahowald, M., Rezk, C.: A resolution of the \({K}(2)\)-local sphere. Ann. Math. 162(2), 777–822 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hazewinkel, M.: Formal groups and applications. Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1978)Google Scholar
  8. 8.
    Henn, H.-W.: On finite resolutions of \({K}(n)\)-local spheres, Elliptic Cohomology. London Mathemetical Society, LMS, Cambridge (2002)Google Scholar
  9. 9.
    Henn, H.-W.: Centralizers of elementary abelian \(p\)-subgroups and mod-\(p\) cohomology of profinite groups. Duke Math. J. 91(3), 561–585 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Karamanov, N.: A propos de la cohomologie du deuxième groupe stabilisateur de Morava; application au calcul de \(\pi _*(L_{K(2)}V(0))\) et du groupe \(Pic_2\) de Hopkins, Ph.D. thesis, Strasbourg (2006)Google Scholar
  11. 11.
    Lazard, M.: Groupes analytiques \(p\)-adiques. Inst. Hautes Études Sci. Publ. Math. 26, 389–603 (1965)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ravenel, D.C.: Complex Cobordism and Stable Homotopy Groups of Spheres, Pure and Applied Mathematics, vol. 121. Academic Press Inc., Orlando (1986)Google Scholar
  13. 13.
    Shimomura, K.: The homotopy groups of the \(L_2\)-localized Toda-Smith spectrum \(V(1)\) at the prime \(3\). Trans. Am. Math. Soc. 349(5), 1821–1850 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shimomura, K.: The homotopy groups of the \(L_2\)-localized mod \(3\) Moore spectrum. J. Math. Soc. Japan 52(1), 65–90 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Strickland, N.P.: Gross-Hopkins duality. Topology 39, 1021–1033 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Hans-Werner Henn
    • 1
    Email author
  • Nasko Karamanov
    • 2
  • Mark Mahowald
    • 3
  1. 1.Institut de Recherche Mathématique Avancée C.N.R.S., Université de Strasbourg StrasbourgFrance
  2. 2.Fakultät für MathematikRuhr-Universität Bochum BochumGermany
  3. 3.Department of MathematicsNorthwestern UniversityEvanstonUSA

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