Mathematische Zeitschrift

, Volume 250, Issue 3, pp 495–513 | Cite as

Alperin’s weight conjecture in terms of equivariant Bredon cohomology

  • Markus Linckelmann
Article

Abstract

Alperin’s weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in §1 and §2, respectively, and §3 provides a tool for computing the cohomology of the functors arising in §2. Taking as starting point the alternating sum formulation of Alperin’s weight conjecture by Knörr-Robinson [11], the material of the previous sections is applied in §4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].

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References

  1. 1.
    Alperin, J.L.: Weights for Finite Groups. Proc. Symp. Pure Math. 47, 1987Google Scholar
  2. 2.
    Alperin, J.L., Broué, M.: Local methods in block theory Ann. of Math. 110, 143–157 (1979)Google Scholar
  3. 3.
    Benson, D. J.: Cohomology of sporadic groups, finite loop spaces, and the Dickson invariants. In: P. H. Kropholler, G. A. Niblo, R. Stöhr, (eds.), Geometry and Cohomology in Group Theory London Math. Soc. Lect. Ser. 252, 1998Google Scholar
  4. 4.
    Boltje, R.: Alperin’s weight conjecture and chain complexes. J. London Math. Soc. 68, 83–101 (2003)CrossRefGoogle Scholar
  5. 5.
    Boltje, R.: Chain complexes for Alperin’s weight conjecture and Dade’s ordinary conjecture in the abelian defect group case. Preprint, to appear: J. Group TheoryGoogle Scholar
  6. 6.
    Broto, C., Levi, R., Oliver, B.: Homotopy equivalences of p-completed classifying spaces of finite groups. Invent. Math. 151, 611–664 (2003)Google Scholar
  7. 7.
    Broto, C., Levi, R., Oliver, B.: The homotopy theory of fusion systems. J. Amer. Math. Soc. 16, 779–856 (2003)CrossRefGoogle Scholar
  8. 8.
    Dwyer, W. G., Kan, D. M.: A classification theorem for diagrams of simplicial sets Topology 23, 139–155 (1984)Google Scholar
  9. 9.
    Grodal, J.: Higher limits via subgroup complexes. Annals of Math. 55, 405–457 (2002)Google Scholar
  10. 10.
    Knörr, R., Robinson, G. R.: Some remarks on a conjecture of Alperin. J. London Math. Soc. (2) 39, 48–60 (1989)Google Scholar
  11. 11.
    Külshammer, B.: Crossed products and blocks with normal defect groups. Commun. Algebra 13, 147–168 (1985)Google Scholar
  12. 12.
    Külshammer,B.: L. Puig, Extensions of nilpotent blocks. Invent. Math. 102, 17–71 (1990)Google Scholar
  13. 13.
    Linckelmann, M.: Fusion category algebras. J. Algebra 277, 222–235 (2004)Google Scholar
  14. 14.
    Lück, W.: Transformation Groups and Algebraic K-Theory. Springer Lecture Notes in Mathematics, Springer Verlag, 1408, 1989Google Scholar
  15. 15.
    Puig, L.: Pointed groups and construction of modules. J. Algebra 116, 7–129 (1988)Google Scholar
  16. 16.
    Robinson, G.R.: Cancellation theorems related to conjectures of Alperin and Dade. J. Algebra 249, 196–219 (2002)Google Scholar
  17. 17.
    Robinson, G.R.: More cancellation theorems related to conjectures of Alperin and Dade. J. Algebra 249, 463–471 (2002)Google Scholar
  18. 18.
    Robinson, G.R.: Weight conjectures for ordinary characters. Preprint, 2003Google Scholar
  19. 19.
    Słomińska, J.: Homotopy colimits on EI-categories. Lecture Notes in Mathematics, Springer Verlag, Berlin 1474, 273–294 (1991)Google Scholar
  20. 20.
    Słomińska, J.: Some spectral sequences in Bredon cohomology. Cahiers Topologie Géom. Différentielle Catégoriques 33, 99–133 (1992)Google Scholar
  21. 21.
    Symonds, P.: The Bredon cohomology of Subgroup Complexes. Preprint, 2002Google Scholar
  22. 22.
    Thévenaz, J.: G-algebras and Modular Representation Theory Oxford Science Publications, Clarendon Press Oxford, 1995Google Scholar
  23. 23.
    Webb, P.J.: A split exact sequence of Mackey functors. Trans. Amer. Math. Soc. 347, 1865–1961 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Markus Linckelmann
    • 1
  1. 1.Ohio State UniversityColumbusUSA

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