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Intermediaries in Bredon (Co)homology and Classifying Spaces

Fotini Dembegioti, Nansen Petrosyan, and Olympia Talelli

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Abstract

For certain contractible $G$-CW-complexes and $\mathfrak F$ a family of subgroups of $G$, we construct a spectral sequence converging to the $\mathfrak F$-Bredon cohomology of $G$ with $\mathrm{E}_1$-terms given by the $\mathfrak F$-Bredon cohomology of the stabilizer subgroups. As applications, we obtain several corollaries concerning the cohomological and geometric dimensions of the classifying space $E_{\mathfrak {F}}G$. We also introduce, for any subgroup closed class of groups $\mathfrak F$, a hierarchically defined class of groups and show that if a group $G$ is in this class, then $G$ has finite $\mathfrak F\cap G$-Bredon (co)homological dimension if and only if $G$ has jump $\mathfrak F\cap G$-Bredon (co)homology.

Article information

Source
Publ. Mat., Volume 56, Number 2 (2012), 393-412.

Dates
First available in Project Euclid: 19 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.pm/1340127811

Mathematical Reviews number (MathSciNet)
MR2978329

Zentralblatt MATH identifier
1326.55007

Subjects
Primary: 20J05: Homological methods in group theory

Keywords
Classifying space Bredon (co)homology spectral sequence

Citation

Dembegioti, Fotini; Petrosyan, Nansen; Talelli, Olympia. Intermediaries in Bredon (Co)homology and Classifying Spaces. Publ. Mat. 56 (2012), no. 2, 393--412. https://projecteuclid.org/euclid.pm/1340127811


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References

  • R. C. Alperin and P. B. Shalen, Linear groups of finite cohomological dimension, Invent. Math. 66(1) (1982), 89\Ndash98. \small\tt DOI: 10.1007/BF01404758.
  • P. Baum, A. Connes, and N. Higson, Classifying space for proper actions and $K$-theory of group $C^{\ast}$-algebras, in: “$C^{\ast}$\guioalgebras: 1943–1993” (San Antonio, TX, 1993), Contemp. Math. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240\Ndash291.
  • D. J. Benson, Complexity and varieties for infinite groups, I, J. Algebra 193(1) (1997), 260\Ndash287. \small\tt DOI: 10.1006/jabr.1996.6996.
  • G. E. Bredon, “Equivariant cohomology theories”, Lecture Notes in Mathematics 34, Springer-Verlag, Berlin-New York, 1967.
  • K. S. Brown, “Cohomology of groups”, Graduate Texts in Mathematics 87, Springer-Verlag, New York-Berlin, 1982.
  • J. F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphism conjectures in $K$- and $L$-theory, $K$-Theory 15(3) (1998), 201\Ndash252. \small\tt DOI: 10.1023/A:1007784106877.
  • R. J. Flores and B. E. A. Nucinkis, On Bredon homology of elementary amenable groups, Proc. Amer. Math. Soc. 135(1) (2007), 5\Ndash11 (electronic). \small\tt DOI: 10.1090/S0002-9939-06-08565-0.
  • M. Fluch, On Bredon (co-)homological dimensions of groups, Ph.D. Thesis, University of Southampton (2011).
  • M. Fluch and B. E. A. Nucinkis, On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups, arXiv:1104.0588v2 (2011).
  • G. Gandini, Cohomological invariants and the classifying space for proper actions, arXiv:1106.3022 (2011).
  • J. A. Hillman, Elementary amenable groups and $4$-manifolds with Euler characteristic $0$, J. Austral. Math. Soc. Ser. A 50(1) (1991), 160\Ndash170.
  • J. H. Jo and B. E. A. Nucinkis, Periodic cohomology and subgroups with bounded Bredon cohomological dimension, Math. Proc. Cambridge Philos. Soc. 144(2) (2008), 329\Ndash336.
  • D. H. Kochloukova, C. Martínez-Pérez, and B. E. A. Nucinkis, Cohomological finiteness conditions in Bredon cohomology,Bull. Lond. Math. Soc. 43(1) (2011), 124\Ndash136. \small\tt DOI: 10.1112/blms/ \small\tt bdq088.
  • P. H. Kropholler, On groups of type $(\mathrm{FP})^{\infty}$, J. Pure Appl. Algebra 90(1) (1993), 55\Ndash67. \small\tt DOI: 10.1016/0022-4049(93)90136-H.
  • P. H. Kropholler, P. A. Linnell, and J. A. Moody, Applications of a new $K$-theoretic theorem to soluble group rings, Proc. Amer. Math. Soc. 104(3) (1988), 675\Ndash684. \small\tt DOI: 10.2307/2046771.
  • W. Lück, “Transformation groups and algebraic $K$-theory”, Lecture Notes in Mathematics 1408, Mathematica Gottingensis, Springer\guioVerlag, Berlin, 1989.
  • W. Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149(2) (2000), 177\Ndash203. \small\tt DOI: 10.1016/S0022-4049(98)90173-6.
  • W. Lück, Survey on classifying spaces for families of subgroups, in: “Infinite groups: geometric, combinatorial and dynamical aspects”, Progr. Math. 248, Birkhäuser, Basel, 2005, pp. 269\Ndash322. \small\tt DOI: 10.1007/3-7643-7447-0$_-$7.
  • W. Lück and M. Weiermann, On the classifying space of the family of virtually cyclic groups, Preprintreihe SFB 478 - Geometrische Strukturen in der Mathematik, Heft 453, Münster.v2, to appear in the Proceedings in honour of Farrell and Jones in Pure and Applied Mathematic Quarterly.
  • C. Martínez-Pérez, A spectral sequence in Bredon (co)ho-mology, J. Pure Appl. Algebra 176(2–3) (2002), 161\Ndash173. \small\tt DOI: 10.1016/\small\tt S0022-4049(02)00154-8.
  • G. Mislin, On the classifying space for proper actions, in: “Cohomological methods in homotopy theory” (Bellaterra, 1998), Progr. Math. 196, Birkhäuser, Basel, 2001, pp. 263\Ndash269.
  • G. Mislin and A. Valette, “Proper group actions and the Baum-Connes conjecture”, Advanced Courses in Mathematics, CRM Barcelona, Birkhüser Verlag, Basel, 2003.
  • B. E. A. Nucinkis, On dimensions in Bredon homology, Homology Homotopy Appl. 6(1) (2004), 33\Ndash47.
  • N. Petrosyan, Jumps in cohomology and free group actions,J. Pure Appl. Algebra 210(3) (2007), 695\Ndash703. \small\tt DOI: 10.1016/ \small\tt j.jpaa.2006.11.011.
  • N. Petrosyan, New action induced nested classes of groups and jump (co)homology, Preprint (2009), arXiv:0911.0541.
  • J.-P. Serre, Cohomologie des groupes discrets, in: “Prospects in mathematics” (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Ann. of Math. Studies 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77\Ndash169.
  • O. Talelli, A characterization of cohomological dimension for a big class of groups, J. Algebra 326 (2011), 238\Ndash244. \small\tt DOI: 10.1016/j.jalgebra.2010.01.021.
  • C. A. Weibel, “An introduction to homological algebra”, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.