Algebraic & Geometric Topology

Nullification functors and the homotopy type of the classifying space for proper bundles

Ramon J Flores

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Abstract

Let G be a discrete group for which the classifying space for proper G–actions is finite-dimensional. We find a space W such that for any such G, the classifying space  B¯G for proper G–bundles has the homotopy type of the W–nullification of  BG. We use this to deduce some results concerning  B¯G and in some cases where there is a good model for  B¯G we obtain information about the  Bp–nullification of  BG.

Article information

Source
Algebr. Geom. Topol., Volume 5, Number 3 (2005), 1141-1172.

Dates
Received: 25 November 2004
Revised: 27 August 2005
Accepted: 29 August 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796447

Digital Object Identifier
doi:10.2140/agt.2005.5.1141

Mathematical Reviews number (MathSciNet)
MR2171806

Zentralblatt MATH identifier
1092.55012

Subjects
Primary: 55P20: Eilenberg-Mac Lane spaces
Secondary: 55P60: Localization and completion

Keywords
(co)localization finite groups Eilenberg–MacLane spaces

Citation

Flores, Ramon J. Nullification functors and the homotopy type of the classifying space for proper bundles. Algebr. Geom. Topol. 5 (2005), no. 3, 1141--1172. doi:10.2140/agt.2005.5.1141. https://projecteuclid.org/euclid.agt/1513796447


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References

  • S I Adyan, Problema Bernsaida i tozhdestva v gruppakh, [The Burnside problem and identities in groups], Izdat. “Nauka”, Moscow (1975)
  • G Z Arone, W G Dwyer, Partition complexes, Tits buildings and symmetric products, Proc. London Math. Soc. (3) 82 (2001) 229–256
  • P Baum, A Connes, N Higson, Classifying space for proper actions and $K$-theory of group $C\sp \ast$-algebras, from: “$C\sp \ast$-algebras: 1943–1993 (San Antonio, TX, 1993)”, Contemp. Math. 167, Amer. Math. Soc. Providence, RI (1994) 240–291
  • A K Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994) 831–873
  • A K Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997) 1321–1354
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer-Verlag, Berlin (1972)
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, New York (1982)
  • W Chachólski, On the functors $CW\sb A$ and $P\sb A$, Duke Math. J. 84 (1996) 599–631
  • J H Conway, The orbifold notation for surface groups, from: “Groups, combinatorics & geometry (Durham, 1990)”, London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press, Cambridge (1992) 438–447
  • H S M Coxeter, W O J Moser, Generators and relations for discrete groups, Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 14, Springer-Verlag, Berlin (1965)
  • W Dicks, P H Kropholler, I J Leary, S Thomas, Classifying spaces for proper actions of locally finite groups, J. Group Theory 5 (2002) 453–480
  • T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. Berlin (1987)
  • E Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer-Verlag, Berlin (1996)
  • M P F du Sautoy, J J McDermott, G C Smith, Zeta functions of crystallographic groups and analytic continuation, Proc. London Math. Soc. (3) 79 (1999) 511–534
  • W G Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997) 783–804
  • W G Dwyer, H-W Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel (2001)
  • R J Flores, Nullification and cellularization of classifying spaces of finite groups, preprint, available at: http://hopf.math.purdue.edu/cgi-bin/generate?/Flores/draft1
  • P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse series 35, Springer-Verlag, New York (1967)
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag, Basel (1999)
  • J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and $E\sb \infty$ structures, Arch. Math. (Basel) 59 (1992) 115–129
  • J F Jardine, Simplicial approximation, Theory Appl. Categ. 12 (2004) No. 2, 34–72
  • A.G. Kurosh, The theory of groups, vol. II, Chelsea publishing Co., 1960. LOOKUP
  • J Lannes, L Schwartz, Sur la structure des $A$-modules instables injectifs, Topology 28 (1989) 153–169
  • I J Leary, B E A Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001) 539–550
  • X Lee, http://www.xahlee.org/Wallpaper_dir/c5_17WallpaperGroups.html
  • S Levy, http://www.geom.umn.edu/docs/reference/CRC-formulas/book.html
  • W Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000) 177–203
  • W Lück, Survey on classifying spaces for families of subgroups, Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik, Heft 308, Münster (2004)
  • W Lück, R Stamm, Computations of $K$- and $L$-theory of cocompact planar groups, $K$-Theory 21 (2000) 249–292
  • S MacLane, Categories for the working mathematician, Springer-Verlag, New York (1971)
  • H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984) 39–87
  • G Mislin, On the classifying space for proper actions, from: “Cohomological methods in homotopy theory (Bellaterra, 1998)”, Progr. Math. 196, Birkhäuser, Basel (2001) 263–269
  • G Mislin, A Valette, Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2003)
  • A Y Ol'shanskiĭ, Geometriya opredelyayushchikh sootnoshenii v gruppakh [The geometry of defining relations in groups], Sovremennaya Algebra [Modern Algebra], “Nauka”, Moscow (1989)
  • D Robinson, Finiteness conditions and generalized soluble groups, Ergebnisse series 62, Springer-Verlag, New York-Berlin (1972)
  • D Schattschneider, The plane symmetry groups: their recognition and notation, Amer. Math. Monthly 85 (1978) 439–450
  • J-P Serre, Cohomologie des groupes discrets, from: “Prospects in mathematics (Proc. Sympos. Princeton Univ. Princeton, NJ, 1970)”, Princeton Univ. Press, Princeton, NJ (1971) 77–169. Ann. of Math. Studies, No. 70
  • R M Switzer, Algebraic topology–-homotopy and homology, Grundlehren series 212, Springer-Verlag, New York (1975)
  • R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91–109