Cellular Covers of Local Groups

  • Ramón Flores
  • Jérôme SchererEmail author


We prove that, in the category of groups, the composition of a cellularization and a localization functor need not be idempotent. This provides a negative answer to a question of Emmanuel Dror Farjoun.

Mathematics Subject Classification

Primary 20F99 Secondary 55P60 20E22 20F50 55R35 



We warmly thank the referee for drawing our attention to the need to define precisely what an idempotent functor actually is and for his kind advice on the terminology we introduce in Sect. 2. We also thank Antonio Viruel, Delaram Kahrobaei and Simon Smith for helpful conversations, and Varujan Atabekyan and Alexander Yu. Ol’shanskiĭ for helping us out with the second homology group of Burnside groups. The first author wishes to thank the École Polytechnique Fédérale de Lausanne for its kind hospitality when this joint project started.


  1. 1.
    A gift to Guido Mislin on the occasion of his retirement from ETHZ, June 2006, Collected by Indira Chatterji. Guido’s book of conjectures, Enseign. Math. (2) 54(1–2), 3–189 (2008)Google Scholar
  2. 2.
    Adian, S.I.: The Burnside problem and identities in groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 95. Springer, Berlin (1979). (Translated from the Russian by John Lennox and James Wiegold) Google Scholar
  3. 3.
    Adian, S.I., Atabekyan, V.: Central extensions of free periodic groups of odd period \(n \ge 665\). Russ. Acad. Sci. Sb. Math. 209, 12 (2018). (English translation to appear in Math. Notes) Google Scholar
  4. 4.
    Blomgren, M., Chachólski, W., Farjoun, E.D., Segev, Y.: Idempotent transformations of finite groups. Adv. Math. 233, 56–86 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure Appl. Algebra 9(2), 207–220 (1976/1977)Google Scholar
  6. 6.
    Casacuberta, C.: Anderson localization from a modern point of view. In: The Čech Centennial (Boston, MA, 1993), Contemporary in Mathematics, vol. 181. American Mathematical Society, Providence, RI, pp. 35–44 (1995)Google Scholar
  7. 7.
    Casacuberta, C., Descheemaeker, A.: Relative group completions. J. Algebra 285(2), 451–469 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chachólski, W.: On the functors \(CW_A\) and \(P_A\). Duke Math. J. 84(3), 599–631 (1996)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dror Farjoun, E.: Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics, vol. 1622. Springer, Berlin (1996)CrossRefGoogle Scholar
  10. 10.
    Dror Farjoun, E., Göbel, R., Segev, Y.: Cellular covers of groups. J. Pure Appl. Algebra 208(1), 61–76 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Flores, R.: Nullification and cellularization of classifying spaces of finite groups. Trans. Am. Math. Soc. 359(4), 1791–1816 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Flores, R.: On the idempotency of some composite functors. Isr. J. Math. 187, 81–91 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Flores, R., Muro, F.: Torsion homology and cellular approximation. Algebraic Geom. Topol. Preprint available. arXiV:1707.07654 (to appear)
  14. 14.
    Hopf, H.: Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv. 14, 257–309 (1942)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Libman, A.: Cardinality and nilpotency of localizations of groups and \(G\)-modules. Isr. J. Math. 117, 221–237 (2000)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Libman, A.: A note on the localization of finite groups. J. Pure Appl. Algebra 148(3), 271–274 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ol’shanskiĭ, Y.A.: Geometry of defining relations in groups. In: Mathematics and its Applications (Soviet Series), vol. 70. Kluwer Academic Publishers Group, Dordrecht (1991) (translated from the 1989 Russian original by Yu. A. Bakhturin) Google Scholar
  18. 18.
    Przeździecki, A.J.: Large localizations of finite groups. J. Algebra 320(12), 4270–4280 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rodríguez, J.L., Scevenels, D.: Iterating series of localization functors. In: Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999), Contemporary in Mathematics, vol. 265. American Mathematical Society, Providence, RI, pp. 211–221 (2000)Google Scholar
  20. 20.
    Rodríguez, J.L., Scherer, J.: Cellular approximations using Moore spaces. In: Cohomological Methods in Homotopy Theory (Bellaterra, 1998), Progress in Mathematics, vol. 196. Birkhäuser, Basel, pp. 357–374 (2001)Google Scholar
  21. 21.
    Rodríguez, J.L., Scherer, J., Strüngmann, L.: On localization of torsion abelian groups. Fundam. Math. 183(2), 123–138 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rodríguez, J.L., Scherer, J., Thévenaz, J.: Finite simple groups and localization. Isr. J. Math. 131, 185–202 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de SevillaSevillaSpain
  2. 2.Institute of Mathematics, EPFL, Station 8LausanneSwitzerland

Personalised recommendations