Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ G$ 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.$

References [Enhancements On Off] (What's this?)

  • [AM94] A. Adem and J.R. Milgram, Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften, vol. 3, Springer-Verlag, 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 part I, Topology 33 (1994), no. 3, 455-492.MR 1286926 (95i:55006)
  • [Asc00] M. Aschbacher, Finite group theory, Cambridge Stud. Adv. Math., vol. 10, Cambridge University Press, 2000. MR 1777008 (2001c:20001)
  • [BC99] J. Berrick and C. Casacuberta, A universal space for plus-constructions, Topology (1999), no. 38, 467-477. MR 1670384 (2000a:55023)
  • [BK72] A.K. Bousfield and D.M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, 1972.MR 0365573 (51:1825)
  • [BL87] R. Brown and J.L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311-335. MR 0899052 (88m:55008)
  • [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)
  • [Bou97] A.K. Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), no. 6, 1321-1354. MR 1481817 (98m:55009)
  • [Bro82] K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, 1982. MR 0672956 (83k:20002)
  • [Cas91] C. Casacuberta, The behaviour of homology in the localization of finite groups, Canad. Math. Bull. 34 (1991), no. 3, 311-320.MR 1127752 (92g:20080)
  • [Cas94] C. Casacuberta, Anderson localization from a modern point of view, The Cech Centennial, Contemp. Math. 181, Amer. Math. Soc., 1994, pp. 35-44. MR 1320986 (96a:55017)
  • [CDI02] W. Chachólski, W.G. Dwyer and M. Intermont, The $ A$-complexity of a space, J. London Math. Soc. 65 (2002), no. 1, 204-222.MR 1875145 (2002j:55009)
  • [Cha96] W. Chachólski, On the functors $ \mathbf{CW}_A$ and $ \mathbf{P}_A$, Duke Math. J. 84 (1996), no. 3, 599-631.MR 1408539 (97i:55023)
  • [DF95] E. Dror-Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Math., vol. 1622, Springer-Verlag, 1995.MR 1392221 (98f:55010)
  • [DGI02] W.G. Dwyer, J. Greenlees and S. Iyengar, Duality in algebra and topology, preprint 2002.
  • [DH00] W.G. Dwyer and H.W. Henn, Homotopy theoretic methods in group cohomology, Adv. Courses Math., vol. 1, CRM Barcelona, 2000. MR 1926776 (2003h:20093)
  • [Dwy94] W.G. Dwyer, The centralizer decomposition of $ \textrm{BG}$, Algebraic topology; New Trends in Localization and Periodicity, Progr. Math., Birkhauser, vol. 136, 1994, 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)
  • [FP78] Z. Fiedorowicz and S. Priddy, Homology of classical groups over finite fields and their associated infinite loop spaces, Lecture Notes in Math., vol. 674, Springer-Verlag, 1978. MR 0513424 (80g:55018)
  • [Gor80] D. Gorenstein, Finite groups, Chelsea Publishing Co., Inc., 1980. MR 0569209 (81b:20002)
  • [JMO92] S. Jackowski, J. McClure and R. Oliver, Homotopy classification of self-maps of $ \textrm{BG}$ via $ \textrm{G}$-actions, Ann. of Math. 135 (1992), no. 1, 183-226. MR 1147962 (93e:55019a)
  • [Kur60] A.G. Kurosh, The theory of groups, vol. II, Chelsea Publishing Co., 1960.MR 0109842 (22:727)
  • [Lev95] R. Levi, On finite groups and homotopy theory, vol. 118, Mem. Amer. Math. Soc., no. 567, November 1995. MR 1308466 (96c:55019)
  • [LS89] J. Lannes and L. Schwartz, Sur la structure des $ \mathcal{A}$-modules instables injectifs, Topology 28 (1989), 153-169.MR 1003580 (90h:55027)
  • [Mil84] H. Miller, On the Sullivan conjecture for mapping spaces, Ann. of Math. (1984), 1-90. MR 0750716 (85i:55012)
  • [MKS76] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Dover Publications, Inc., 1976. 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)
  • [Nei94] J.A. Neisendorfer. Localization and connected covers of finite complexes, The Cech Centennial, Contemp. Math. 181, Amer. Math. Soc., 1994, pp. 385-390. MR 1321002 (96a:55019)
  • [Rob96] D.J. Robinson, A course in the theory of groups, Grad. Texts in Math., vol. 80, Springer-Verlag, 1996. MR 1357169 (96f:20001)
  • [RS98] J.L. Rodríguez and J. Scherer, Cellular approximations using Moore spaces, Cohomological methods in homotopy theory, Progr. in Math, vol. 196, 1998, pp. 357-374. MR 1851263 (2003a:55024)
  • [RS00] J.L. Rodríguez and D. Scevenels, Iterating series of localization functors, Une Dégustation Topologique: Homotopy Theory in the Swiss Alps. Contemp. Math. 265, 2000, pp. 211-221.MR 1804680 (2001k:55031)
  • [Sch94] L. Schwartz, Unstable modules over the Steenrod algebra and Sullivan fixed point set conjecture, Chicago University Press, 1994.MR 1282727 (95d:55017)
  • [Tho86] C.B. Thomas, Characteristic classes and the cohomology of finite groups, Cambridge Stud. Adv. Math., vol. 9, Cambridge University Press, 1986.MR 0878978 (88f:20005)
  • [Wei77] M. Weinstein, Examples of groups, Polygonal Publishing House, Passaic, N.J., 1977.MR 0453847 (56:12100)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 55P20, 55P60

Retrieve articles in all journals with MSC (2000): 55P20, 55P60

Additional Information

Ramón J. Flores
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Barcelona, E–08193 Bellaterra, Spain

Keywords: (Co)localization, finite groups, Eilenberg-MacLane spaces.
Received by editor(s): January 27, 2005
Published electronically: October 17, 2006
Additional Notes: This work was partially supported by MCYT grant BFM2001-2035.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society