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Cellular Covers of Local Groups

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Abstract

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.

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References

  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)

  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. 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. Blomgren, M., Chachólski, W., Farjoun, E.D., Segev, Y.: Idempotent transformations of finite groups. Adv. Math. 233, 56–86 (2013)

    Article  MathSciNet  Google Scholar 

  5. Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure Appl. Algebra 9(2), 207–220 (1976/1977)

  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)

  7. Casacuberta, C., Descheemaeker, A.: Relative group completions. J. Algebra 285(2), 451–469 (2005)

    Article  MathSciNet  Google Scholar 

  8. Chachólski, W.: On the functors \(CW_A\) and \(P_A\). Duke Math. J. 84(3), 599–631 (1996)

    Article  MathSciNet  Google Scholar 

  9. Dror Farjoun, E.: Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Mathematics, vol. 1622. Springer, Berlin (1996)

    Book  Google Scholar 

  10. Dror Farjoun, E., Göbel, R., Segev, Y.: Cellular covers of groups. J. Pure Appl. Algebra 208(1), 61–76 (2007)

    Article  MathSciNet  Google Scholar 

  11. Flores, R.: Nullification and cellularization of classifying spaces of finite groups. Trans. Am. Math. Soc. 359(4), 1791–1816 (2007)

    Article  MathSciNet  Google Scholar 

  12. Flores, R.: On the idempotency of some composite functors. Isr. J. Math. 187, 81–91 (2012)

    Article  MathSciNet  Google Scholar 

  13. Flores, R., Muro, F.: Torsion homology and cellular approximation. Algebraic Geom. Topol. Preprint available. arXiV:1707.07654 (to appear)

  14. Hopf, H.: Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv. 14, 257–309 (1942)

    Article  MathSciNet  Google Scholar 

  15. Libman, A.: Cardinality and nilpotency of localizations of groups and \(G\)-modules. Isr. J. Math. 117, 221–237 (2000)

    Article  MathSciNet  Google Scholar 

  16. Libman, A.: A note on the localization of finite groups. J. Pure Appl. Algebra 148(3), 271–274 (2000)

    Article  MathSciNet  Google Scholar 

  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)

  18. Przeździecki, A.J.: Large localizations of finite groups. J. Algebra 320(12), 4270–4280 (2008)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  21. Rodríguez, J.L., Scherer, J., Strüngmann, L.: On localization of torsion abelian groups. Fundam. Math. 183(2), 123–138 (2004)

    Article  MathSciNet  Google Scholar 

  22. Rodríguez, J.L., Scherer, J., Thévenaz, J.: Finite simple groups and localization. Isr. J. Math. 131, 185–202 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Sevilla, 41080, Sevilla, Spain

    Ramón Flores

  2. Institute of Mathematics, EPFL, Station 8, 1015, Lausanne, Switzerland

    Jérôme Scherer

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  1. Ramón Flores
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  2. Jérôme Scherer
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Correspondence to Jérôme Scherer.

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This work was supported by the Spanish Ministry of Education and Science MEC-FEDER Grant MTM2010-15831 and MTM2016-80439-P, and the Andalusian government under Grants FQM-213 and P07-FQM-2863.

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Flores, R., Scherer, J. Cellular Covers of Local Groups. Mediterr. J. Math. 15, 229 (2018). https://doi.org/10.1007/s00009-018-1273-y

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