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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.
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
55P60 Localization and completion in homotopy theory
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[1] S I Adyan, Problema Bernsaida i tozhdestva v gruppakh, Izdat. “Nauka”, Moscow (1975) 335 · Zbl 0306.20045
[2] G Z Arone, W G Dwyer, Partition complexes, Tits buildings and symmetric products, Proc. London Math. Soc. \((3)\) 82 (2001) 229 · Zbl 1028.55008
[3] 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
[4] A K Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994) 831 · Zbl 0839.55008
[5] A K Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997) 1321 · Zbl 0886.55011
[6] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer (1972) · Zbl 0259.55004
[7] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036
[8] W Chachólski, On the functors \(CW_A\) and \(P_A\), Duke Math. J. 84 (1996) 599 · Zbl 0873.55014
[9] J H Conway, The orbifold notation for surface groups, London Math. Soc. Lecture Note Ser. 165, Cambridge Univ. Press (1992) 438 · Zbl 0835.20048
[10] 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
[11] 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
[12] T tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, Walter de Gruyter & Co. (1987) · Zbl 0611.57002
[13] E D Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer (1996) · Zbl 0842.55001
[14] 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
[15] W G Dwyer, Homology decompositions for classifying spaces of finite groups, Topology 36 (1997) 783 · Zbl 0872.55014
[16] W G Dwyer, H W Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag (2001) · Zbl 1047.55001
[17] R J Flores, Nullification and cellularization of classifying spaces of finite groups, Trans. Amer. Math. Soc. 359 (2007) 1791 · Zbl 1112.55009
[18] 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
[19] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag (1999) · Zbl 0949.55001
[20] 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
[21] J F Jardine, Simplicial approximation, Theory Appl. Categ. 12 (2004) 34 · Zbl 1062.55019
[22] A G Kurosh, The theory of groups, Chelsea Publishing Co. (1960) · Zbl 0094.24501
[23] J Lannes, L Schwartz, Sur la structure des \(A\)-modules instables injectifs, Topology 28 (1989) 153 · Zbl 0683.55016
[24] I J Leary, B E A Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology 40 (2001) 539 · Zbl 0983.55010
[25] X Lee, The 17 wallpaper groups
[26] S Levy, Geometry formulas and facts
[27] W Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000) 177 · Zbl 0955.55009
[28] W Lück, Survey on classifying spaces for families of subgroups, Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik, Heft 308, Münster (2004)
[29] W Lück, R Stamm, Computations of \(K\)- and \(L\)-theory of cocompact planar groups, \(K\)-Theory 21 (2000) 249 · Zbl 0979.19003
[30] S MacLane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1971) · Zbl 0705.18001
[31] H Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. \((2)\) 120 (1984) 39 · Zbl 0552.55014
[32] G Mislin, On the classifying space for proper actions, Progr. Math. 196, Birkhäuser (2001) 263 · Zbl 1001.55013
[33] G Mislin, A Valette, Proper group actions and the Baum-Connes conjecture, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag (2003) · Zbl 1028.46001
[34] A Y Ol’shanskiĭ, Geometriya opredelyayushchikh sootnoshenii v gruppakh, \cyrSovremennaya Algebra., “Nauka” (1989) 448
[35] D J S Robinson, Finiteness conditions and generalized soluble groups I, Ergebnisse der Mathematik und ihrer Grenzgebiete 62, Springer (1972) · Zbl 0243.20032
[36] D Schattschneider, The plane symmetry groups: their recognition and notation, Amer. Math. Monthly 85 (1978) 439 · Zbl 0381.20036
[37] J P Serre, Cohomologie des groupes discrets, Princeton Univ. Press (1971) · Zbl 0229.57016
[38] R M Switzer, Algebraic topology-homotopy and homology, Die Grundlehren der mathematischen Wissenschaften 212, Springer (1975) · Zbl 0305.55001
[39] 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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