Classifying Spaces for Wallpaper Groups

  • Ramón J. Flores
Conference paper
Part of the Trends in Mathematics book series (TM)


In this paper we use the homotopy structure of the classifying space for proper bundles of symmetries group of the plane to describe the Bℤ/p-nullification, in the sense of Dror-Farjoun, of the classifying spaces of these groups.


Fundamental Group Middle Point Homotopy Type Orbit Space Klein Bottle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BCH94]
    P. Baum, A. Connes and N. Higson. Classifying space for proper actions and K-Theory of group C*-Algebras Contemp. Math, 167:241–291, 1994.MathSciNetGoogle Scholar
  2. [BF03]
    A.J. Berrick and E. Dror-Farjoun Fibrations and nullifications. Israel J. Math., 135:205–220, 2003.MATHMathSciNetGoogle Scholar
  3. [BK72]
    A.K. Bousfield and D.M. Kan. Homotopy limits, completions and localizations. Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.MATHGoogle Scholar
  4. [Bou94]
    A.K. Bousfield. Localization and periodicity in unstable homotopy theory. J. Amer. Math. Soc., 7(4):831–873, 1994.MATHCrossRefMathSciNetGoogle Scholar
  5. [CCS05]
    N. Castellana, J.A. Crespo and J. Scherer. Deconstructing Hopf spaces. Preprint 2004, available at Scholar
  6. [Cha96]
    W. Chachólski. On the functors CW A and P A. Duke Math. J., 84(3):599–631, 1996.MATHCrossRefMathSciNetGoogle Scholar
  7. [CM65]
    H. Coxeter and W. Moser. Generators and relations for discrete groups, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 2nd edition, Berlin, 1965.Google Scholar
  8. [Die87]
    T. tom Dieck. Transformation groups, volume 8 of de Gruyter Studies in Mathematics. de Gruyter and Co., 1987.Google Scholar
  9. [DMG99]
    J. Dermott, M. du Sautoy and G. Smith. Zeta functions of crystallographic groups and analytic continuation. Proc. Lond. Math. Soc., 79(3):511–534, 1999.CrossRefGoogle Scholar
  10. [Dwy96]
    W.G. Dwyer. The centralizer decomposition of BG. In Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994), volume 136 of Progr. Math., pages 167–184. Birkhäuser, Basel, 1996.Google Scholar
  11. [Far96]
    E. Dror Farjoun. Cellular spaces, null spaces and homotopy localization, volume 1622 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996.Google Scholar
  12. [FL02]
    F.A. Farris and R. Lanning. Wallpaper functions. Expo. Math., 20(3):193–214, 2002.MATHMathSciNetGoogle Scholar
  13. [Flo04]
    R.J. Flores. Nullification and cellularization of classifying spaces of finite groups. To appear in Trans. Amer. Math. Soc. Google Scholar
  14. [Flo05]
    R.J. Flores. Nullification functors and the homotopy type of the classifying space for proper actions. Algebr. Geom. Topol., 5(46):1141–1172. 2005.MATHCrossRefMathSciNetGoogle Scholar
  15. [Joy05]
    D.E. Joyce. Scholar
  16. [Las56]
    R. Lashof. Classification of fibre bundles by the fibre space of the base. Ann. of Math., 64(3):436–446, 1956.CrossRefMathSciNetGoogle Scholar
  17. [LN01]
    I.J. Leary and B.E.A. Nucinkis. Every CW-complex is a classifying space for proper bundles. Topology, 40(3):539–550, 2001.MATHCrossRefMathSciNetGoogle Scholar
  18. [Lee05]
    X. Lee. Scholar
  19. [LS00]
    W. Lück and R. Stamm. Computations of K-and L-theory of cocompact planar groups. K-theory, 21(3):242–292, 2000.CrossRefGoogle Scholar
  20. [Lüc89]
    W. Lück. Transformation groups and algebraic K-theory, volume 1408 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1989.Google Scholar
  21. [Lüc00]
    W. Lück. The type of the classifying space for a family of subgroups. J. Pure Appl. Algebra, 149(2):177–203, 2000.MATHCrossRefMathSciNetGoogle Scholar
  22. [Lüc05]
    W. Lück. Survey on classifying spaces for families of subgroups. Preprint, available at Scholar
  23. [MT89]
    M. Mahowald and R. Thompson. K-Theory and unstable homotopy groups, volume 96 of Contemp. Math. Amer. Math. Soc., 1989.Google Scholar
  24. [Mil84]
    H. Miller. The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2), 120(1):39–87, 1984.CrossRefMathSciNetGoogle Scholar
  25. [Mis03]
    G. Mislin. Equivariant K-homology of the classifying space for proper actions. In Proper group actions and the Baum-Connes conjecture (Bellaterra, 2001), volume 3 of Adv. Cours. Math. CRM, pages 1–78. Birkhäuser, Basel, 2003.Google Scholar
  26. [RS01]
    J.L. Rodríguez and J. Scherer. Cellular approximations using Moore spaces. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 357–374. Birkhäuser, Basel, 2001.Google Scholar
  27. [Sc78]
    D. Schattschneider. The plane symmetry groups: their recognition and notation. Amer. Math. Monthly, 85(6):439–450, 1978.MATHCrossRefMathSciNetGoogle Scholar
  28. [Ser71]
    J.P. Serre. Cohomologie des groupes discrets. In Prospects in Mathematics, Proc. Sympos. Princeton Univ. (Princeton, N.J., 1970), volume 70 of Ann. of Math. Studies., pages 77–169. Princeton University Press, 1971.MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Ramón J. Flores
    • 1
  1. 1.Departamento de EstadísticaUniversidad Carlos IIIColmenarejo, MadridSpain

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