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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Subgroup posets, Bredon cohomology and equivariant Euler characteristics


Author: Conchita Martínez-Pérez
Journal: Trans. Amer. Math. Soc. 365 (2013), 4351-4370
MSC (2010): Primary 20J05, 18G35, 18G30, 20J06
Published electronically: February 7, 2013
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Abstract: For discrete groups $ \Gamma $ with a bound on the order of their finite subgroups, we construct Bredon projective resolutions of the trivial module in terms of projective covers of the chain complex associated to the poset of finite subgroups. We use this to give new results on dimensions of $ \underline {\operatorname {E}}\Gamma $ and to reprove that for virtually solvable groups, $ \underline {\mathrm {cd}}\Gamma =\mathrm {vcd}\Gamma $. We also deduce a formula to compute the equivariant Euler class of $ \underline {\operatorname {E}}\Gamma $ for $ \Gamma $ virtually solvable of type $ \mathrm {FP}_\infty $ and use it to compute orbifold Euler characteristics.


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

  • 1. A. Adem, M. Klaus. Lectures on orbifolds and group cohomology, Advanced Lectures in Mathematics 16, Transformation Groups and Moduli Spaces of Curves. Higher Education Press, edited by L.Ji and S-T Yau, 2010.
  • 2. A. Adem, J. Leida, Y. Ruan. Orbifolds and string topology. Cambridge Tracts in Mathematics, Vol. 171, Cambridge Un. Press, 2007. MR 2359514 (2009a:57044)
  • 3. M. F. Atiyah, G. Segal. On equivariant Euler characteristics. J. Geom. Physics 6 (4), (1989), 671-677. MR 1076708 (92c:19005)
  • 4. D. J. Benson.
    Representations and cohomology I: Basic representation theory of finite groups and associative algebras.
    Cambridge University Press, Cambridge 1991. MR 1110581 (92m:20005)
  • 5. K. S. Brown.
    Cohomology of groups.
    Springer-Verlag, New York 1982. MR 672956 (83k:20002)
  • 6. K. S. Brown. Complete Euler characteristics and fixed-point theory. J. of Pure and Appl. Algebra 24 (1982), 103-121. MR 651839 (83d:20035)
  • 7. F. Connolly and T. Kozniewski.
    Finiteness properties of classifying spaces of proper $ \gamma $-actions.
    J. Pure Appl. Algebra 41 (1), (1986), 17-36. MR 844461 (87j:57026)
  • 8. J. Cornick and P. H. Kropholler.
    Homological finiteness conditions for modules over group algebras.
    J. London Math. Soc. 58 (2), (1998), 49-62. MR 1666074 (99k:20105)
  • 9. B. Eckmann. Amenable groups and Euler characteristic. Comment. Math. Helv. 67 (1992), 383-393. MR 1171301 (93k:57070)
  • 10. P. Kropholler, C. Martínez-Pérez, and B. E. A. Nucinkis. Cohomological finiteness conditions for elementary amenable groups. J. Reine Angew. Math. 637 (2009), 49-62. MR 2599081 (2011e:20074)
  • 11. P. H. Kropholler and G. Mislin.
    Groups acting on finite-dimensional spaces with finite stabilizers.
    Comment. Math. Helv. 73 (1), (1998), 122-136. MR 1610595 (99f:20086)
  • 12. P. Kropholler and O. Talelli.
    On a property of fundamental groups of graphs of finite groups.
    J. Pure Appl. Algebra 74, (1991), 57-59. MR 1129129 (92h:57003)
  • 13. M. Langer, W. Lück. On the group cohomology of the semi-direct product $ \mathbb{Z}^n\rtimes _\rho \mathbb{Z}_m$ and a conjecture of Adem-Ge-Petrosyan. http://arxiv.org/pdf/1105.4772v1
  • 14. I. J. Leary and B. E. A. Nucinkis. Some groups of type VF. Invent. Math. 151 (1) (2003), 135-165. MR 1943744 (2003k:20086)
  • 15. J. C. Lennox and D. J. S. Robinson.
    The Theory of Infinite Soluble Groups.
    Oxford Un. Press, New York, 2004. MR 2093872 (2006b:20047)
  • 16. W. Lück.
    Transformation groups and algebraic $ K$-theory, volume 1408 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1989. MR 1027600 (91g:57036)
  • 17. W. Lück.
    The type of the classifying space for a family of subgroups.
    J. Pure Appl. Algebra 149 (2), (2000), 177-203. MR 1757730 (2001i:55018)
  • 18. W. Lück.
    $ L^2$-Invariants: Theory and Applications to Geometry and K-Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer, 2002.
  • 19. C. Martínez-Pérez.
    A bound for the Bredon cohomological dimension.
    J. Group Theory 10 (6), (2007), 731-747. MR 2364823 (2008j:20159)
  • 20. C. Martínez-Pérez and B. E. A. Nucinkis. Virtually soluble groups of type $ \mathrm {FP}_\infty $.
    Comment. Math. Helv. 85 (1), (2010), 135-150. MR 2563683 (2011e:20075)
  • 21. P. Symonds.
    The Bredon cohomology of subgroup complexes.
    J. Pure Appl. Algebra 199(1-3), (2005), 261-298. MR 2134305 (2006e:20093)
  • 22. J. Thevenaz. Equivariant $ K$-theory and Alperin's conjecture. J. Pure Appl. Algebra 85 (2), (1993), 185-202. MR 1207508 (94c:20022)
  • 23. Ch. A. Weibel.
    An introduction to homological algebra,
    Cambridge University Press, Cambridge 1994. MR 1269324 (95f:18001)

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Additional Information

Conchita Martínez-Pérez
Affiliation: Departamento de Matemáticas, IUMA. Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: conmar@unizar.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05781-9
PII: S 0002-9947(2013)05781-9
Keywords: Bredon cohomology, virtually soluble group, proper classifying space, poset of finite subgroups
Received by editor(s): October 17, 2011
Received by editor(s) in revised form: December 19, 2011
Published electronically: February 7, 2013
Additional Notes: The author was partially supported by BFM2010-19938-C03-03, Gobierno de Aragón and the European Union’s ERDF funds
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.