A Mini-Course on Morava Stabilizer Groups and Their Cohomology

  • Hans-Werner HennEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2194)


These notes are slightly edited notes of a mini-course of 4 lectures delivered at the Vietnam Institute for Advanced Study in Mathematics in August 2013. The aim of the course was to introduce participants to joint work of the author with Goerss, Karamanov, Mahowald and Rezk which uses group cohomology in a crucial way to give a new approach to previous work by Miller et al. (Ann. Math. 106:469–516, 1977), and by Shimomura and his collaborators (Shimomura, J. Math. Soc. Jpn. 52(1):65–90, 2000; Shimomura, Topology 41(6):1183–1198, 2002; Shimomura and Yabe, Topology 34(2):261–289, 1995). This new approach has lead to a better understanding of old results as well as to substantial new results.


  1. [Bea15]
    A. Beaudry, The chromatic splitting conjecture at n = p = 2. arXiv:1502.02190v2Google Scholar
  2. [Bou79]
    A.K. Bousfield, The localization of spectra with respect to homology. Topology 18(4), 257–281 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [DH04]
    E.S. Devinatz, M.J. Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43(1), 1–47 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [GH04]
    P. Goerss, M. Hopkins, Moduli spaces of commutative ring spectra, Structured Ring Spectra. London Mathematical Society. Lecture Note Series, vol. 315 (Cambridge University Press, Cambridge, 2004), pp. 151–200Google Scholar
  5. [GH16]
    P. Goerss, H.-W. Henn, The Brown-Comenetz dual of the K(2)-local sphere at the prime 3. Adv. Math. 288, 648–678 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [GHM04]
    P. Goerss, H.-W. Henn, M. Mahowald, The homotopy of L 2 V (1) for the prime 3, in Categorical Decomposition Techniques in Algebraic Topology (Isle of Skye, 2001). Progress in Mathematics, vol. 213 (Birkhäuser, Basel, 2004), pp. 125–151Google Scholar
  7. [GHM14]
    P.G. Goerss, H.-W. Henn, M. Mahowald, The rational homotopy of the K(2)-local sphere and the chromatic splitting conjecture for the prime 3 and level 2. Doc. Math. 19, 1271–1290 (2014)MathSciNetzbMATHGoogle Scholar
  8. [GHMR05]
    P. Goerss, H.-W. Henn, M. Mahowald, C. Rezk, A resolution of the K(2)-local sphere at the prime 3. Ann. Math. (2) 162(2), 777–822 (2005)Google Scholar
  9. [GHMR15]
    P. Goerss, H.-W. Henn, M. Mahowald, C. Rezk, On Hopkins’ Picard groups for the prime 3 and chromatic level 2. J. Topol. 8(1), 267–294 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Hen07]
    H.-W. Henn, On finite resolutions of K(n)-local spheres, in Elliptic Cohomology. London Mathematical Society Lecture Note Series, vol. 342 (Cambridge University Press, Cambridge, 2007), pp. 122–169Google Scholar
  11. [Hen98]
    H.-W. Henn, Centralizers of elementary abelian p-subgroups and mod-p cohomology of profinite groups. Duke Math. J. 91(3), 561–585 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [HHR16]
    M.A. Hill, M.J. Hopkins, D.C. Ravenel, On the nonexistence of elements of Kervaire invariant one. Ann. Math. 184, 1–262 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [HKM13]
    H.-W. Henn, N. Karamanov, M. Mahowald, The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited. Math. Z. 275(3–4), 953–1004 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [HS99]
    M. Hovey, N.P. Strickland, Morava K-Theories and Localisation. Memoirs of the American Mathematical Society, vol. 139(666) (American Mathematical Society, Providence, RI, 1999)Google Scholar
  15. [Kar06]
    N. Karamanov, À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π L K(2) V (0) et du Pic2 de Hopkins. Ph.D. thesis, Université Louis Pasteur (2006)Google Scholar
  16. [Kar10]
    N. Karamanov, On Hopkins’ Picard group Pic2 at the prime 3. Algebr. Geom. Topol. 10(1), 275–292 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Lad13]
    O. Lader, Une résolution projective pour le seconde groupe de Morava pourp ≥ 5 et applications. Ph. D. thesis, Université de Strasbourg (2013). Google Scholar
  18. [Laz65]
    M. Lazard, Groupes analytiques p-adiques. Inst. Hautes Études Sci. Publ. Math. 26, 389–603 (1965)MathSciNetzbMATHGoogle Scholar
  19. [Mil81]
    H. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space. J. Pure Appl. Algebra 20(3), 287–312 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [MRW77]
    H.R. Miller, D.C. Ravenel, S.W. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence. Ann. Math. 106, 469–516 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [NSW08]
    J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 323 (Springer, Berlin, 2008)Google Scholar
  22. [Rav84]
    D.C. Ravenel, Localization with respect to certain periodic homology theories. Am. J. Math. 106(2), 351–414 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Rav86]
    D.C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, vol. 121 (Academic, Orlando, FL, 1986)Google Scholar
  24. [Ser71]
    J.-P. Serre, Cohomologie des groupes discrets. Ann. Math. Stud. 70, 77–169 (1971)MathSciNetzbMATHGoogle Scholar
  25. [Shi00]
    K. Shimomura, The homotopy groups of the L 2-localized mod 3 Moore spectrum. J. Math. Soc. Jpn. 52(1), 65–90 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [SW02]
    K. Shimomura, X. Wang, The homotopy groups π (L 2 S 0) at the prime 3. Topology 41(6), 1183–1198 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [SY95]
    K. Shimomura, A. Yabe, The homotopy groups π (L 2 S 0). Topology 34(2), 261–289 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Sym07]
    P. Symonds, Permutation complexes for profinite groups. Comment. Math. Helv. 82(1), 1–37 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeC.N.R.S. - Université de StrasbourgStrasbourgFrance

Personalised recommendations