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Nullification functors and the homotopy type of the classifying space for proper bundles. (English) Zbl 1092.55012
Given two spaces $$A$$ and $$X$$, $$X$$ is $$A$$-null if the mapping space $$\text{map}(A,X)$$ is homotopy equivalent to $$X$$ via the inclusion of the constant maps, $$X\hookrightarrow \text{map}(A,X)$$. The $$A$$-nullification of a space is a functor on spaces that assigns to $$X$$ an $$A$$-null space $$P_A(X)$$ together with a universal map $$X\to P_A(X)$$ which induces a weak homotopy equivalence $$\text{map}(P_A(X),Y) \simeq \text{map}(X,Y)$$ for every $$A$$-null space $$Y$$. Thus $$P_A$$ ‘makes trivial the structure of $$X$$ that depends on $$A$$.’
Let $$G$$ denote a discrete group throughout. The paper relates nullification with respect to $$W_\infty = \bigvee_p B\mathbb{Z}_p$$, where the index is over the prime numbers, and the classifying space for proper actions of a group $$G$$, denoted $$\underline{\text{B}}G$$. The space $$\underline{\text{B}}G$$ enjoys the universal property that if $$G$$ acts on $$X$$ with isotropy groups all finite, then there is a $$G$$-map $$X\to \underline{\text{E}}G$$ unique up to $$G$$ homotopy. The space $$\underline{\text{B}}G = \underline{\text{E}}G/G$$. The main theorem shows that $$P_{W_\infty}(BG) \simeq \underline{\text{B}}G$$.
Based on this equivalence, the author explores the properties of the classifying space for proper actions. As a functor $$\underline{\text{B}}$$ takes products to products, and certain pushouts of groups to the corresponding pushout of classifying spaces. Using the properties of $$\bigvee_p B\mathbb{Z}_p$$ and nullification, there are results for the homotopy type of $$\underline{\text{B}}G$$ for $$G$$ locally finite, supersoluble, or one of the wallpaper groups. Since $$\underline{\text{B}}G$$ and $$BG$$ are realized as realizations of nerves of small categories, the analysis of $$BG \to \underline{\text{B}}G$$ can be studied by simplicial means, and the author determines the homotopy fibre of this map to end the paper.
##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55P60 Localization and completion in homotopy theory
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##### References:
  S I Adyan, Problema Bernsaida i tozhdestva v gruppakh, Izdat. “Nauka”, Moscow (1975) 335 · Zbl 0306.20045  G Z Arone, W G Dwyer, Partition complexes, Tits buildings and symmetric products, Proc. London Math. Soc. $$(3)$$ 82 (2001) 229 · Zbl 1028.55008  P Baum, A Connes, N Higson, Classifying space for proper actions and $$K$$-theory of group $$C^*$$-algebras, Contemp. Math. 167, Amer. Math. Soc. (1994) 240 · Zbl 0830.46061  A K Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994) 831 · Zbl 0839.55008  A K Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997) 1321 · Zbl 0886.55011  A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972) · Zbl 0259.55004  K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036  W Chachólski, On the functors $$CW_A$$ and $$P_A$$, Duke Math. J. 84 (1996) 599 · Zbl 0873.55014  J H Conway, The orbifold notation for surface groups, London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press (1992) 438 · Zbl 0835.20048  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 (1965) · Zbl 0133.28002  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 · Zbl 1060.20035  T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. (1987) · Zbl 0611.57002  E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996) · Zbl 0842.55001  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 · Zbl 1039.11056  W G Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997) 783 · Zbl 0872.55014  W G Dwyer, H W Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag (2001) · Zbl 1047.55001  R J Flores, Nullification and cellularization of classifying spaces of finite groups, Trans. Amer. Math. Soc. 359 (2007) 1791 · Zbl 1112.55009  P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer New York, New York (1967) · Zbl 0186.56802  P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag (1999) · Zbl 0949.55001  J Hollender, R M Vogt, Modules of topological spaces, applications to homotopy limits and $$E_\infty$$ structures, Arch. Math. $$($$Basel$$)$$ 59 (1992) 115 · Zbl 0766.55006  J F Jardine, Simplicial approximation, Theory Appl. Categ. 12 (2004) 34 · Zbl 1062.55019  A G Kurosh, The theory of groups, Chelsea Publishing Co. (1960) · Zbl 0094.24501  J Lannes, L Schwartz, Sur la structure des $$A$$-modules instables injectifs, Topology 28 (1989) 153 · Zbl 0683.55016  I J Leary, B E A Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001) 539 · Zbl 0983.55010  X Lee, The 17 wallpaper groups  S Levy, Geometry formulas and facts  W Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000) 177 · Zbl 0955.55009  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 · Zbl 0979.19003  S MacLane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1971) · Zbl 0705.18001  H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. $$(2)$$ 120 (1984) 39 · Zbl 0552.55014  G Mislin, On the classifying space for proper actions, Progr. Math. 196, Birkhäuser (2001) 263 · Zbl 1001.55013  G Mislin, A Valette, Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag (2003) · Zbl 1028.46001  A Y Ol’shanskiĭ, Geometriya opredelyayushchikh sootnoshenii v gruppakh, \cyrSovremennaya Algebra., “Nauka” (1989) 448  D J S Robinson, Finiteness conditions and generalized soluble groups I, Ergebnisse der Mathematik und ihrer Grenzgebiete 62, Springer (1972) · Zbl 0243.20032  D Schattschneider, The plane symmetry groups: their recognition and notation, Amer. Math. Monthly 85 (1978) 439 · Zbl 0381.20036  J P Serre, Cohomologie des groupes discrets, Princeton Univ. Press (1971) · Zbl 0229.57016  R M Switzer, Algebraic topology-homotopy and homology, Die Grundlehren der mathematischen Wissenschaften 212, Springer (1975) · Zbl 0305.55001  R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 · Zbl 0392.18001
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