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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
Article

Abstract

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 \(TMF\).

Notes

Acknowledgments

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.

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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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