Nullification and cellularization of classifying spaces of finite groups

Flores, Ramón

doi:10.1090/S0002-9947-06-03926-2

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ISSN 1088-6850(online) ISSN 0002-9947(print)

Nullification and cellularization of classifying spaces of finite groups

Author: Ramón J. Flores
Journal: Trans. Amer. Math. Soc. 359 (2007), 1791-1816
MSC (2000): Primary 55P20, 55P60
Published electronically: October 17, 2006
MathSciNet review: 2272149
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Abstract: In this note we discuss the effect of the $\textrm{B}\mathbb{Z}/p$ -nullification $\mathbf{P}_{\textrm{B}\mathbb{Z}/p}$ and the $\textrm{B}\mathbb{Z}/p$ -cellularization $\mathbf{CW}_{\textrm{B}\mathbb{Z}/p}$ over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe $\mathbf{P}_{\textrm{B}\mathbb{Z}/p}\textrm{B}G$ by means of a covering fibration, and we classify all finite groups for which $\textrm{B}G$ is $\textrm{B}\mathbb{Z}/p$ -cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of $\mathbf{P}_{\textrm{B}\mathbb{Z}/p}\textrm{B}G$ and $\mathbf{CW}_{\textrm{B}\mathbb{Z}/p}\textrm{B}G.$

[AM94] Alejandro Adem and R. James Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer-Verlag, Berlin, 1994. MR 1317096 (96f:20082)
[ABN94] J. Aguadé, C. Broto, and D. Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke. I, Topology 33 (1994), no. 3, 455–492. MR 1286926 (95i:55006), http://dx.doi.org/10.1016/0040-9383(94)90023-X
[Asc00] M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008 (2001c:20001)
[BC99] A. J. Berrick and Carles Casacuberta, A universal space for plus-constructions, Topology 38 (1999), no. 3, 467–477. MR 1670384 (2000a:55023), http://dx.doi.org/10.1016/S0040-9383(97)00073-6
[BK72] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573 (51 #1825)
[BL87] Ronald Brown and Jean-Louis Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335. With an appendix by M. Zisman. MR 899052 (88m:55008), http://dx.doi.org/10.1016/0040-9383(87)90004-8
[Bou94] A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831–873. MR 1257059 (95c:55010), http://dx.doi.org/10.1090/S0894-0347-1994-1257059-7
[Bou97] A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), no. 6, 1321–1354. MR 1481817 (98m:55009)
[Bro82] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956 (83k:20002)
[Cas91] Carles Casacuberta, The behaviour of homology in the localization of finite groups, Canad. Math. Bull. 34 (1991), no. 3, 311–320. MR 1127752 (92g:20080), http://dx.doi.org/10.4153/CMB-1991-051-5
[Cas94] Carles Casacuberta, Anderson localization from a modern point of view, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 35–44. MR 1320986 (96a:55017), http://dx.doi.org/10.1090/conm/181/02028
[CDI02] W. Chachólski, W. G. Dwyer, and M. Intermont, The 𝐴-complexity of a space, J. London Math. Soc. (2) 65 (2002), no. 1, 204–222. MR 1875145 (2002j:55009), http://dx.doi.org/10.1112/S0024610701002691
[Cha96] Wojciech Chachólski, On the functors 𝐶𝑊_{𝐴} and 𝑃_{𝐴}, Duke Math. J. 84 (1996), no. 3, 599–631. MR 1408539 (97i:55023), http://dx.doi.org/10.1215/S0012-7094-96-08419-7
[DF95] Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. MR 1392221 (98f:55010)
[DGI02] W.G. Dwyer, J. Greenlees and S. Iyengar, Duality in algebra and topology, preprint 2002.
[DH00] William G. Dwyer and Hans-Werner Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2001. MR 1926776 (2003h:20093)
[Dwy94] W. G. Dwyer, The centralizer decomposition of 𝐵𝐺, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 167–184. MR 1397728 (97i:55028)
[Dwy97] W. G. Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997), no. 4, 783–804. MR 1432421 (97m:55016), http://dx.doi.org/10.1016/S0040-9383(96)00031-6
[FP78] Zbigniew Fiedorowicz and Stewart Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Mathematics, vol. 674, Springer, Berlin, 1978. MR 513424 (80g:55018)
[Gor80] Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209 (81b:20002)
[JMO92] Stefan Jackowski, James McClure, and Bob Oliver, Homotopy classification of self-maps of 𝐵𝐺 via 𝐺-actions. I, Ann. of Math. (2) 135 (1992), no. 1, 183–226. MR 1147962 (93e:55019a), http://dx.doi.org/10.2307/2946568
[Kur60] A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842 (22 #727)
[Lev95] Ran Levi, On finite groups and homotopy theory, Mem. Amer. Math. Soc. 118 (1995), no. 567, xiv+100. MR 1308466 (96c:55019)
[LS89] Jean Lannes and Lionel Schwartz, Sur la structure des 𝐴-modules instables injectifs, Topology 28 (1989), no. 2, 153–169 (French). MR 1003580 (90h:55027), http://dx.doi.org/10.1016/0040-9383(89)90018-9
[Mil84] Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716 (85i:55012), http://dx.doi.org/10.2307/2007071
[MKS76] Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434 (54 #10423)
[MP01] G. Mislin and G. Peschke, Central extensions and generalized plus-constructions, Trans. Amer. Math. Soc. 353 (2001), no. 2, 585–608. MR 1804509 (2002f:19003), http://dx.doi.org/10.1090/S0002-9947-00-02727-6
[Nei94] Joseph A. Neisendorfer, Localization and connected covers of finite complexes, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 385–390. MR 1321002 (96a:55019), http://dx.doi.org/10.1090/conm/181/02044
[Rob96] Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169 (96f:20001)
[RS98] José L. Rodríguez and Jérôme Scherer, Cellular approximations using Moore spaces, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 357–374. MR 1851263 (2003a:55024)
[RS00] José L. Rodríguez and Dirk Scevenels, Iterating series of localization functors, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999) Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp. 211–221. MR 1804680 (2001k:55031), http://dx.doi.org/10.1090/conm/265/04251
[Sch94] Lionel Schwartz, Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1282727 (95d:55017)
[Tho86] C. B. Thomas, Characteristic classes and the cohomology of finite groups, Cambridge Studies in Advanced Mathematics, vol. 9, Cambridge University Press, Cambridge, 1986. MR 878978 (88f:20005)
[Wei77] Michael Weinstein, Examples of groups, Polygonal Publishing House, Passaic, N.J., 1977. MR 0453847 (56 #12100)